SB    Sh3    3m 


MECHANICS 


MOLECULAR  PHYSICS  AND  HEAT 


A  TWELVE  WEEKS'   COLLEdE  COURSE 


BY 


ROBERT  ANDREWS   MILLIKAN,  Pn.D. 

M 

ASSISTANT  PROFESSOR  OF  PHYSICS  IN  THK 
UNIVERSITY  OF  CHICAGO 


BOSTON,  U.S.A. 

GINN    &   COMPANY,   PUBLISHERS 
1903 


COPYRIGHT,  1902 
BY  SCOTT,  FORESMAN  &  COMPANY 


COPYRIGHT,  1903 
BY  GINN  &  COMPANY 


ALL    RIGHTS   RESERVED 


!.**•  c« .' 

««;%*•*  •«•" 


PEEFACE 

This  book  is  neither  a  laboratory  manual  in  the  ordinary  sense 
of  the  term,  nor  yet  is  it  simply  a  class-room  text.  It  is  intended 
to  take  the  place  of  both.  It  represents  the  first  portion  of  a 
college  course  in  General  Physics  in  which  the  primary  object  has 
been  to  establish  an  immediate  and  vital  connection  between  theory 
and  experiment.  Of  course  such  connection  always  exists  in  the 
mind  of  the  teacher;  but  the  use  in  class-room  and  laboratory  of 
separate  texts,  separate  courses,  and  separate  instructors  is  on  the 
whole  unfavorable  to  making  it  clear  to  the  student.  The  stu- 
dent who  takes  an  experimental  course  which  is  out  of  imme- 
diate connection  with  class-room  discussion,  who  is  provided  in 
the  laboratory  with  an  isolated  set  of  directions,  or  with  a  labora- 
tory manual  which  is  essentially  a  compendium  of  directions  for 
all  conceivable  experiments,  may  perhaps  in  some  cases  obtain, 
with  the  aid  of  references  to  text-books,  a  comprehensive  grasp  of 
the  theory  and  bearings  of  his  experiment;  but  it  is  safe  to  say 
that  in  a  great  majority  of  cases  he  does  not  do  so.  The  most 
serious  criticism  which  can  be  urged  against  modern  laboratory 
work  in  Physics  is  that  it  often  degenerates  into  a  servile  following 
of  directions,  and  thus  loses  all  save  a  purely  manipulative  value. 
Important  as  is  dexterity  in  the  handling  and  adjustment  of 
apparatus,  it  can  not  be  too  strongly  emphasized  that  it  is  grasp  of 
principles,  not  skill  in  manipulation  which  should  be  the  primary 
object  of  General  Physics  courses. 

Furthermore,  an  intimate  connection  between  lecture  and 
laboratory  work  is  no  less  important  from  the  standpoint  of  the 
former  than  of  the  latter.  Without  the  fixing  power  of  laboratory 
applications,  a  thorough  grasp  of  physical  principles  is  seldom,  or 
never,  gained.  This  is  particularly  true  in  Mechanics,  the  most 
fundamental  of  all  the  branches  of  Physics,  for  it  is  only  through 
it  that  the  door  is  opened  to  insight  into  the  theories  of  Heat, 
Sound,  Light,  and  Electricity. 

3 


4  PREFACE 

In  the  second  place,  this  book  represents  an  attempt  to  teach 
thoroughly  a  few  fundamental  principles  rather  than  to  present 
superficially  a  large  mass  of  facts.  Most  of  the  general  texts 
which  combine  a  full  presentation  of  facts,  with  a  satisfactory 
discussion  of  their  relation  to  theory,  have  grown  too  bulky  for 
general  class-room  use.  On  the  other  hand,  other  texts  have 
appeared  in  which  the  necessity  for  condensation  has  resulted  in 
such  abridgment  of  discussion  that  there  is  little  left  but  a 
skeleton  of  experimental  and  theoretical  results. 

In  attempting  to  avoid  both  of  these  extremes,  a  selection  lias 
been  made  for  the  present  course  of  such  principles  only  as  can  be 
most  effectively  presented  in  connection  with  laboratory  demon- 
strations. This  course  is  presented  in  the  first  third  (twelve 
weeks)  of  a  year  of  General  Physics  in  the  Junior  College  at  the 
University  of  Chicago.  The  time  is  divided  nearly  equally  between 
class-room  and  laboratory  work ;  but  the  former  is  wholly  occupied 
with  the  discussion  and  application  to  practical  problems,  of  the 
twenty-three  principles  presented  in  the  text.  No  demonstration 
lectures  whatever  are  given.  The  second  third  of  the  college  year 
is  occupied  with  the  presentation,  by  the  same  general  plan,  of 
those  parts  of  Electricity,  Light,  and  Sound  which  can  be  most 
profitably  studied  in  connection  with  laboratory  instruments  and 
methods.  This  course  is  embodied  in  a  second  volume.  The 
last  third  of  the  year  is  devoted  wholly  to  demonstration  lectures 
upon  subjects  which  have  been  omitted  from  the  preceding  courses 
because  they  are  more  suitable  to  lecture  than  to  laboratory 
methods  of  presentation.  Such  are,  for  example,  Static  Elec- 
tricity, Electric  Radiation,  the  Discharge  of  Electricity  through 
Gases,  the  Radiation,  Absorption,  Polarization,  and  Interfer- 
ence of  Light,  Physiological  Optics  and  Acoustics;  in  a  word, 
all  phenomena  the  presentation  of  which  requires  primarily  quali- 
tative rather  than  quantitative  experiment.  This  demonstration 
lecture  course  is  given  last  instead  of  first  because  a  thorough 
grounding  in  the  fundamental  principles  of  physical  measurement 
is  deemed  absolutely  essential  to  the  intelligent  following  of  lec- 
tures of  college  grade  in  any  branch  of  Physics. 

A  third  aim  has  been  to  prepare  a  book  which  should  represent 
a  continuous  course  in  the  study  of  the  principles  of  experimental 
Physics,  rather  than  a  compendium  either  of  facto  or  experiments. 


PREFACE  5 

Hence,  in  the  first  place,  all  purely  manipulative  experiments 
have  been  altogether  omitted.  A  student  is  not  required  to  make 
a  useless  measurement  for  the  sake  of  learning  to  use  an  instru- 
ment ;  he  rather  learns  to  use  the  instrument  for  the  sake  of  making 
a  needed  measurement.  The  inversion  of  this  order  has  invariably 
weakened  interest  in  laboratory  work.  In  the  second  place,  all 
needless  repetition  of  slight  variations  of  the  same  experiment  has 
been  avoided.  For  example,  in  the  subject  of  Heat,  there  is  but 
one  general  principle  involved  in  the  method  of  mixtures,  whether 
it  be  applied  to  the  determination  of  the  specific  heat  of  a  solid,  or 
of  a  liquid,  the  latent  heat  of  fusion,  or  of  vaporization.  Hence 
it  has  been  illustrated  in  this  course  by  but  one  laboratory  exercise. 
Similarly,  the  usual  half-dozen  or  more  experiments  upon  the  den- 
sities of  liquids  and  solids  have  been  reduced  to  two.  In  a  word, 
experiments  have  been  made  incidental  to  the  study  of  principles, 
not  principles  incidental  to  the  study  of  experiments. 

Fourthly,  an  especial  effort  has  been  made  to  present  Physics 
as  a  science  of  exact  measurement.  Much  harm  is  often  done  by 
attempting  to  teach  a  course  in  an  exact  science  with  the  aid  of 
instruments  capable  of  giving  results  which  can  be  called  quanti- 
tative by  courtesy  only,  and  which  therefore  foster  the  impression 
that  science  is  after  all  very  inexact.  The  apparatus  which  is 
used  in  this  course  has  therefore  been  selected  and  designed  with 
a  special  reference  to  its  ability  to  yield  accurate  results  in  the 
hands  of  average  students.  All  of  the  new  pieces,  such  as  the 
acceleration  machine  (p.  11),  the  model  balance  (p.  36),  the  bal- 
listic pendulums  (pp.  54  and  62),  the  Young's  modulus  (p.  67), 
the  torsion  machines  (p.  75),  the  inertia  disc  (p.  82),  the 
pendulum  arrangement  (p.  97),  the  centripetal  machine  (p.  102), 
the  pressure  apparatus  (p.  117),  the  form  of  air  thermometer 
(p.  131),  the  vapor  pressure  arrangement  (p.  158),  have  been 
designed  in  this  laboratory,  slight  modifications  having  in  some 
instances  been  introduced  by  the  instrument  maker,  William 
Gaertner,  from  whom  any  of  the  special  pieces  used  in  the  course 
may  be  obtained.  Furthermore,  in  nearly  all  of  the  exercises,  the 
quantity  sought  is  obtained  by  two  distinct  methods  and  the 
results  compared. 

Finally,  since  the  book  represents  a  college,  not  a  high  school, 
course,  the  aim  has  been,  not  so  much  to  acquaint  the  student 


6  PREFACE 

with  interesting  and  striking  phenomena,  as  to  give  him  an  insight 
into  the  real  significance  of  physical  things — to  introduce  him  to 
the  very  heart  of  the  subject  by  putting  him  in  touch  with  the 
methods  and  instruments  of  modern  physical  investigation,  and  by 
carrying  him  through  the  processes  of  close  reasoning  by  which 
the  present  science  of  Physics  has  been  developed.  Students  who 
enter  upon  the  course  are  expected  both  to  have  completed  a  year 
of  secondary  school  Physics,  and  to  have  gained  some  familiarity 
with  the  principles  of  Trigonometry. 

The  author's  justification  for  the  publication  of  this,  the  first 
book  in  the  series,  is  the  hope  that  the  presentation  of  a  method 
of  instruction  which  has  been  found  most  satisfactory  in  the  Uni- 
versity of  Chicago  may  not  be  altogether  without  usefulness,  or  at 
least  suggestiveness,  to  teachers  of  Physics  in  other  institutions. 
The  custom  usually  followed  is  to  hold  lectures  and  quizzes  upon 
a  group  of  eight  experiments,  before  taking  up  the  laboratory 
work.  The  problems  are  of  course  invaluable  aids  in  the  fixing 
of  principles.  In  the  laboratory  not  more  than  twenty  students 
are  ever  permitted  in  a  single  section.  No  more  than  two  dupli- 
cate pieces  of  apparatus  are  ever  used.  Duplicates  of  the  forms  of 
record  which  have  been  inserted  in  the  manual  are  filled  out  by 
the  student  and  handed  to  the  instructor  as  each  experiment  is 
completed.  In  addition  to  filling  out  these  record  slips  each 
student  keeps  a  systematic  note-book,  in  which  are  entered  in 
the  laboratory,  not  at  home,  all  observations  and  all  calcula- 
tions of  whatever  kind.  This  note-book  is  a  complete  record  of 
the  student's  work,  and  should  of  course  be  so  arranged  as  to 
be  easily  intelligible  to  anyona,  even  though  he  be  unfamiliar  with 
the  course. 

This  book  is  the  successor  of  A  College  Course  of  Laboratory 
Experiments  in  General  Physics,  published  by  the  present  director 
of  the  National  Bureau  of  Standards,  Professor  S.  W.  Stratton, 
and  the  author.  It  is  to  Professor  Stratton  that  the  design  of 
much  of  the  apparatus  is  due.  In  the  preparation  of  the  present 
course  the  author  has  had  the  invaluable  assistance  not  only  of 
Professor  Stratton,  but  also  of  Mr.  G.  M.  Hobbs  and  Dr.  H.  G0 
Gale,  both  instructors  in  Physics  at  the  University  of  Chicago, 

UNIVERSITY  OF  CHICAGO, 
August  27,  1902. 


CONTENTS 

MECHANICS 


PAGE 

I.  UNIFORMLY  ACCELERATED  MOTION  ..,'„.  9 
II.  FORCE  PROPORTIONAL  TO  RATE  OF  CHANGE  OF  MOMENTUM 

(f  =  ma]    .        .        .      ';       ...        .      \                •."       .  15 

III.  COMPOSITION  AND  RESOLUTION  OF  FORCES       .        .        .  21 

IV.  THE  PRINCIPLE  OF  WORK     .        .        .        .        .        .*••'.•  29 

V.  ENERGY  AND  EFFICIENCY  .        .        .       .       .       .        .  42 

VI.  THE  LAWS  OF  IMPACT    .        .        ,        .        ,        .        .        .  52 

VII.  ELASTIC  IMPACT.     COEFFICIENT  OF  RESTITUTION    .        .  58 

VIII.   ELASTICITY.    HOOKE'S  LAW:  YOUNG'S  MODULUS      .       .  65 

IX.  THE  COEFFICIENT  OF  RIGIDITY  .        .                        .        .  71 

X.  MOMENT  OF  INERTIA       .-       .        .        .        .        .        .        .78 

XL  SIMPLE  HARMONIC  MOTION         .        .        .        .  :;    .        .  87 

XII.  DETERMINATION  OF  "g" .        •        •        •        •        •        •        •  95 

XIII.  THE  LAW  OF  CENTRIPETAL  FORCE  .         .        .  100 


MOLECULAR    PHYSICS    AND    HEAT 

XIV.  BOYLE'S  LAW  .        .        .        .        .        .        .        .        .  >        105 

XV.  DENSITY  OF  AIR.        ,        .[    '  .        .        .        .    .  ..  .           114 

XVI.  THE  MEASUREMENT  OF  TEMPERATURE  .        ...  .        122 

XVII.  LAW  OF  AVOGADRO — DENSITIES  OF  GASES  AND  VAPORS  138 
XVIII.  THE   PRESSURE-TEMPERATURE  CURVE   OF    A   SATURATED 

VAPOR       .        .        ...       . .        .        .        .  .        152 

XIX.  HYGROMETRY        .        .       .        .        .        .    "  (       .  .           164 

XX.  ARCHIMEDES'  PRINCIPLE        .        .        .        .        .        .  ..       173 

XXI.  CAPILLARITY        .        ...        .        .        .        .  .           181 

XXII.  CALORIMETRY   .        .        .        .        .        .        .        ."-     ,  .        193 

XXIII.  EXPANSION    .        .        .        .        .       .        .        .      ^  .           215 

APPENDIX      .        .        .        .        .        .       .        .        .     '  •        .  .        223 

INDEX 

7 


MECHANICS ;    *  \ 


I 

UNIFORMLY  ACCELERATED    MOTION 

Theory 

Conceive  of  a  body  moving  in  a  straight  line  with  continually 

changing  velocity.     Its  motion  is  said  to  be  uniformly  accelerated 

when  it  makes  equal  gains  of  velocity  in  equal  intervals 

acceleration    of  time.     The  rate  of  change  of  velocitv,  or,  for  the 

defined.  .  .        .    , . 

present  case,  the  gain  in  velocity  per  unit  01  time  is 
called  the  acceleration  of  the  body. 

LAWS  OF  UNIFORMLY  ACCELERATED  MOTION. — The  following 
laws  are  derived  at  once  from  the  above  definition : 

1.  If  v  represent  the  velocity  of  the  body  at  the  end  of  t  units 
of  time,  a  its  acceleration  and  VQ  its  velocity  at  the  beginning  of 

the  t  units,  then 
Velocity  in 

terms  of  *.  _  nf  ,   „,  .  /i  \ 

acceleration  "  V°>  V J  > 

or,  if  the  body  start  from  rest  (i.e.  if  v0  =  0) 

v  =  at.  (2) 

This  law  is  nothing  more  than  the  mathematical  statement  of  the 
definition. 

2.  If  st,  s2,  s3y  5n,  and  sn  +  1  represent  the  distances  traversed 
AcceUra-        ^v  the  body  during  the  1st,  2d,  3d,  nth  and  ( 
o/°8w?ce?ms   un^s  °f  ^me  respectively,  then 

sive  apace  /o  \ 

intervals.  .  «  =  S2—  5i  =  S3  — 52  = Sn  +  i  —  Sa  (O) 


±: 


FIGURE  1 


10  MECHANICS 

Proof.  —  Let  the  straight  line  (Fig.  1)  be  the  path  of  a  body 
moving  with  uniform  acceleration,  and  let  v^  vz,  etc.  ,  be  the  veloc- 
ities at  the  ends  of  the  units  of  time  1,  2,  etc.,  and  s^  s2>  etc.,  the 
spaces  traversed  during  these  units.  The  'space  passed  over  during 
any,in£eVval  Of  time  must  always  be  the  mean  velocity  multiplied 
by  'the  number,  of  units  of  time  in  the  interval.  In  case  the 
velocity.  increases  uniformly  this  mean  velocity  is  evidently  the 
half  sum  of  the  velocities  at  the  beginning  and  at  the  end  of  the 
interval.  Hence,  e.g.  (See  Fig.  1), 

.  (4) 


Similarly  s3  =  vz  +  -r-     .'.  s±  —s3  =  v3  -  vz.     But  v3  -  vz  is  by  defi- 

/o 

nition  a.     .*.  s4  —  s3  =  a.     Similarly  for  s3  —  sa,  etc.         Q.  E.  D. 

The  acceleration  can  therefore  be  most  directly  determined  by 
measuring  distances  traversed  in  successive  units  of  time. 

3.  If  &  represent  the  total  space  traversed  during  t  units  of 

Space  in  time>  then 

%%$**  tf-iof  +  t,.*;  (5) 

or,  if  the  body  start  from  rest 

S=.%at\  (6) 

Proof.  —  Total  space  =  mean  velocity  x  time  =  4  (initial  velocity 

+  final  velocity)  x  time  =  y«+(^  +  gO  x  t  =  ±  at*  +  ^.      Q.  E.  D. 

Z 

4.  If  v  represent  the  velocity  of  a  body  after  it  has  moved 
over  a  space  8  with  an  acceleration  «,  and  if   v0  represent  the 

velocity  which  the  body  had  at  the  point  from  which 
X?o/  in      S  is  measured,  then 

space  and  .  -  -  -  - 

accelera-  V=V%aS+V<f.  (7) 

or,  if  the  body  start  from  rest,  (i.e.,  if  VQ-  0) 

v  =  N/2otf.  (8) 

Proof.  —  From  (1),  v  =  at  +  v0  .  '.  t  =  —    —•  Now  if  v  be  used  to 
represent  the  average  velocity  with  which  the  body  traverses  the 

a     ,  ,  -        V   +  VQ     ,  a  V   +  VQ         V   -  VQ         V*    —  V<?          _ 

space  8,  then  v  =  —  —  -»  but  S  =  vt  =  —  ^~  x  -     —  =  —  -  —  —    Hence 

Z  A  OL  Ztt 

solving  for  v, 


v  =  v/2a£  +  vf ;  or  if  v0  =0,v=  \/2aS.  Q.  E.  D. 


UNIFORMLY    ACCELERATED    MOTION 


11 


Experiment 

The  object  of  this  experiment  is  to  study  the  motion  of  a 
freely  falling  body;  in  particular,  (1)  to  ascertain  whether  a  freely 
falling  body  moves  with  uniformly  accelerated  motion, 
and  (2)  to  determine  the  acceleration  of  this  motion 
in  centimeters  per  second. 

The  falling  body  is  a  brass  frame  «,  (Fig.  2),  which  weighs 
about    1  kgm.   and  falls   a   dis- 
tance of  some  120  cm.  through 
guides  which  offer  very 

srssusx. little  fricti°n-   Tbe 

cord,  which  is  shown 
in  the  figure  passing  over  the  pul- 
ley and  supporting  balancing 
weights,  is  in  this  experiment  de- 
tached from  the  frame  «,  so  that 
the  latter  falls  freely.  The  fall- 
ing frame  carries  with  it  a  tuning 
fork  #,  one  prong  of  which  is  pro- 
vided with  a  light  stylus  which, 
during  the  descent,  traces  a  wavy 
line  upon  the  blackened  glass 
plate  M  (see  Fig.  3).  The  frame 
is  first  raised  to  the  top  of  the 
ways  where  it  is  held  in  place  by 
an  eccentric  catch,  which  keeps 
the  prongs  slightly  spread.  Turn- 
ing the  lever  I  draws  the  eccentric 
up  into  the  crossbar  o,  releases 
the  prongs,  and  allows  the  frame 
and  vibrating  fork  to  fall.  The 
plate  M  can  be  shifted  sidewise  so 
that  a  number  of  traces  can  te 
obtained.  Two  dash-pots  at  the 
base  of  the  instrument  catch  the 
falling  frame  and  take  up  the  jar. 
DIRECTIONS. — Hang  a  plumb- 
bob  from  r  and  adjust  the  level- 
FIGUBE  2  ing  screws  in  the  base  until  a  FIGURE  3 


12  MECHANICS 

strictly  vertical  fall  is  assured.  Remove  the  glass  plate  M  and 
smoke  it  evenly  from  top  to  bottom  in  a  gum-camphor  flame. 

Replace  it  and  adjust  the  stylus  so  that  it  will  follow 
ondmeoStre-  smo°thty  behind  the  falling  fork,  pressing  very  lightly 
tface°f  the  against  tne  P^te.  Then  release  the  fork.  Having 

obtained  at  least  two  good  traces,  remove  the  plate,  set 
it  in  the  horizontal  wooden  frame  provided  (see  Fig.  4),  and  count 
off  the  waves  by  tens  from  top  to  bottom  of  each  trace,  marking 


FIGURE  4 


as  accurately  as  possible  the  crest  of  ea-ch  tenth  wave  by  means  of 
a  pin-scratch  (see  Fig.  3).  Next  turn  the  plate  over,  smoked  side 
down,  and  hold  a  meter  stick  on  edge  upon  the  unsmoked  side  of 
the  plate  just  over  the  trace.  It  will  then  be  easy  to  read  upon 
the  scale  the  distances  between  successive  pin-marks. 

Although  the  scale  and  the  trace  are  the  thickness  of  the  glass 
apart,  the  error  of  parallax,  i.  e.,  the  error  in  reading  arising 
because  the  line  of  sight  is  not  perpendicular  to  the 
P^Q,  may  be  avoided  entirely.  For,  since  the 
blackened  plate  acts  as  a  mirror,  it  is  only  necessary, 
in  taking  a  reading,  to  place  the  eye  so  that  the  images  of  the 
scale  divisions  and  the  scale  divisions  themselves  appear  in  the 
same  straight  line.  In  taking  the  readings  estimate  in  each  case 
to  tenths  mm. 

By  subtracting  Si  from  %   #2  from  s3,  s3  from   s4,   etc.    (see 

Fig.  3),  obtain  from  each  .trace  as  many  values  of  the  acceleration 

as  is  possible,  when  the  unit  of  time  chosen  is  the  time 

Acceleration  , , 

for  different  of  10  vibrations  of  the  fork.  Repeat  the  same  opera- 
tion when  the  unit  is  the  time  of  20  vibrations;  then 
when  it  is  the  time  of  30  vibrations ;  then  of  40.  If  the  motion 
is  uniformly  accelerated  the  values  of  a  will  not  change  as  you 
move  from  the  top  to  the  bottom  of  the  plate. 


UNIFORMLY    ACCELERATED    MOTION 


13 


By  comparison  of  the  mean  values  of  the  accelerations  for  differ- 
ent  units  of   time   find  the   law  which   connects  the 
numerical  value  of  the  acceleration  with  the  length  of 
'value  of  g.     the  unit  of  time.     From  this  law  and  the  rate  of  the 
fork,  as  furnished  by  the  instructor,  determine  #,  the 
acceleration  per  second  of  the  falling  body. 

The  true  value  of  g  is  980  cm.  On  account  of  the  slight  fric- 
tion in  the  guides  in  addition  to  the  air  resistance,  the  value  here 
found  will  fall  a  trifle  short.  From  the  total  length  of 
resistance  the  fall  and  the  correct  value  of  g  'calculate  what  should 
be  the  total  time  of  descent  of  the  falling  body.  Com- 
pare this  with  the  observed  time  obtained  from  the  rate  of  the  fork 
and  the  total  number  of  waves  upon  the  plate.  The  difference 
represents  the  retardation  due  to  the  friction  of  the  air  and  of  the 
guides.  The  difference  between  980  and  the  observed  value  of  g 
represents  of  course  the  negative  acceleration  due  to  friction. 
Since,  when  force  is  measured  in  dynes,  force  =  mass  x  accelera- 
tion (see  Ex.  II),  it  is  possible  from  an  observation  of  the  mass  of 
the  frame  and  fork  to  express  the  value  of  the  retarding  frictional 
force  in  dynes. 

Record 


FIRST  TRACE 

10        20        30       40 
vib's.  vib's.  vib's.  vib's. 


Mean 

Means  reduced  * 
Mean  of  means 
Rate  of  fork  =  — 


vib.  per  sec.  .•.  g  = 


SECOND  TRACE 

10        20        30        40 
vib's.  vib's.  vib's.  vib's. 


Mean  of  means 

•'•  9—  ~ 

%  of  difference  between  the  two  values  of  g  (observational  error)  =  - 
%  of  departure  of  final  mean  from  correct  value  of  g  =  .  .  .  - 
Total  time  of  fall,  calc'd  value  =  —  obs'd  value  =—  %  dif.  =  - 
Weight  of  frame  and  fork  =  —  .'.  No.  dynes  friction  =  - 

*  Divide  mean  of  20's  by  4,  of  30's  by  9,  of  40's  by  16. 


14  MECHANICS 

Problems 

1.  Show  that  the  law  discovered  experimentally  above,  viz., 
that  the  numerical  value  of  the  acceleration  varies  directly  as  the 
square  of  the  unit  of  time,  can  also  be  obtained  theoretically  from 
the  definition  of  acceleration,  viz.,  the  velocity  (measured  in  cm. 
per  second)  which  is  gained  during  one  second. 

2.  If   the   earth's   period   of    rotation   were    to   be   suddenly 
doubled,  and  if  a  second  of  time  were  still  to  be  defined  as  the 
same  fraction  of  that  period  as  at  present,  what  effect  would  be 
observed  in  the  numerical  value  of  g  9 

Disregard  all  centrifugal  tendencies. 

3.  If  the  earth's  period  were  slowly  changing,  how  could  the 
change  be  detected? 

4.  Show  that  a  body  shot  from  the  earth  upward  has  at  any 
given  height  the  same  velocity  in  the  ascent  as  in  the  descent. 

5.  The  Eiffel  Tower  is  335  meters  high.     How  many  seconds 
will  elapse  before  an  arrow,  shot  upward  from  the  tower  with  a 
velocity  of  4000  cm.  per  sec.  reaches  the  earth? 


II 

FORCE  PROPORTIONAL  TO  RATE  OF  CHANGE 
OF   MOMENTUM  (f=ma) 

Theory 

NEWTON'S  LAWS  OF  MOTION. — The   whole  of   Mechanics  is 
based  upon  three  great  experimental  principles  first  clearly  stated 
in  Newton's  laws  of  motion.     As  will  presently  appear, 
o}°iawsnt       the  First  and  Second  Laws  contain  but  one  experimental 
principle,  the  Third  with  the  subjoined  scholium  con- 
tains two.     Newton's  statement  of  these  laws  is -as  follows : 

/.  Every  body  continues  in  its  state  of  rest  or  of  uniform 
motion  in  a  straight  line  except  in  so  far  as  it  is  compelled  by  force 
to  change  that  state. 

II.  (Rate  of)  change  of  (quantity  of)  motion  is  proportional 
to  force  and  takes  place  in  the  straight  line  in  which  the  force  acts. 

III.  To  every  action  there  is  ahvays  an  equal  and  contrary 
reaction;  or  the  mutual  actions  of  any  two  bodies  are  always  equal 
and  oppositely  directed. 

The  First  Law  is  completely  contained  in  the  Galilean  defini- 
tion of  force,  viz. :    "Force  is  that  which  changes  the 
tfeFir*tLaw   s^e  °f  res^  or  uniform  motion  of  a  body."*     It  is 
therefore  essentially  a  definition  of  force,  and  hence 
requires  no  proof. 

The  Second  Law  asserts  that  force  is  proportional  to,  and  there- 
fore may  be  measured  by,  the  rate  of  change  of  quantity  of  motion. 
But  "quantity  of  motion,""  commonly  called  momentum, 
¥wmd*Law.   'IS  ^e  Pr°duct  of  two  factors,  mass  and  velocity.     Rate 
of  change  of  momentum    is  therefore   mass  x  rate  of 

*  Since  the  existence  of  forces  in  equilibrium  is  denied  by  this  defini- 
tion of  force  it  is  often  found  thus  modified.  "Force  is  that  which 
changes  or  tends  to  change,"  etc.  Newton,  however,  (and  in  this  most 
modern  writers  follow  him)  regarded  equilibrium  as  the  balancing  of 
motions,  not  of  forces.  Thus  according  to  the  Newtonian  view  each  force 
actually  produces  its  proper  motion  which  is  neutralized  by  the  equal  and 
opposite  motion  produced  by  the  opposing  force. 

IS 


16  MECHANICS 

change  of  velocity,  or  mass  x  acceleration.  The  mathematical 
statement  of  the  Second  Law  is  therefore/  oc  ma.  This  law  is  a 
definition  of  the  measure  of  a  force;  but  a  little  con- 
sideration will  show  that  it  is  a  definition  which 
presupposes  experimental  knowledge,  and  conse- 
quently finds  its  justification  in  experiment  only. 

Thus,   suppose  two  bodies  whose  masses  are  m1 
and  m2  (see  Fig.  5)  are  dropped  together  toward  the 

earth.     The  fact  that  they  fall  shows  that 
Second^Law.   certain   forces   act  upon  them  (see  First 

Law).     Let  these  forces  be  denoted  by/ 
and  /2.     Now  observation  shows  that  the  two  bodies 
fall  at  the  same  rate,  i.  e.,  that  they  have 
a  common  acceleration.     Call  this  accel-   ^      [—  -  ]       |        |      f~ 
eration  g.    Then  the  assertion  made  in  the    m2     m2         m3          m3 
statement  of  the  Second  Law  is  that  FIGURE  5i 


/2 

Next  suppose  that  the  two  bodies  are  brought  back  to  their 
original  positions,  where,  necessarily,  the  same  two  original  forces, 
/!  and/g,  act  upon  them.  Let  now  each  body  descend  again,  but 
this  time  let  each  be  obliged  to  drag  along  with  it  some  third  mass 
which  would  otherwise  remain  at  rest.  Thus  let  ml  be  placed  upon 
either  one  of  the  two  equal  masses  m^  which  hang  over  a  pulley 
of  negligible  weight  and  friction  (see  Fig.  5).  In  this  case  the 
moving  force  is  evidently  /,  the  mass  moved  is  ml  +  2w3,  and 
the  acceleration  is  some  measurable  quantity  a^.  Then  let  the 
masses  w3  be  brought  back  to  their  original  positions  and  m% 
placed  upon  one  of  them.  The  moving  force  is  now  /2,  the  mass 
moved  mz  +  2w3,  and  the  acceleration  some  measurable  quantity, 
#2.  The  assertion  made  by  the  Second  Law  in  regard  to  these  two 
motions  is 

/' 


.  (10) 

/2 


Now  (9)  and  (10)  cannot  both  be  true  unless 

flt  , 


Since  there  is  no  sort  of  a  priori  ground  for  asserting  the  correct- 


NEWTON'S  SECOND  LAW  17 

ness  of  this  relation,  it  is  only  the  experimental  proof  of  it  which 
furnishes  the  justification  for  the  Second  Law.  This  proof  can  be 
made  in  any  laboratory.  But  the  most  convincing  evidence  of  the 
absolute  correctness  of  the  Second  and  Third  Laws  is  furnished 
by  astronomical  observations.  Predictions  of  eclipses  made  years 
in  advance  are  wholly  based  upon  these  laws  taken  in  connection 
with  the  law  of  gravitation. 

UNITS  OF  FORCE. — If  we  arbitrarily  choose  as  the  absolute 
unit  of  force  the  force  which  is  required  to  impart  to  unit  mass 
one  unit  of  acceleration  the  expression  /  a  ma  changes 
*o/  =  ma 5  ^or  ^»  ^y  definition,  /=  1  when  m  =  1  and 
a  =  1,  then  evidently  /=  4  when  m  =  2  and  a  =  2,  etc. 
The  equation  f=ma  is  then  merely  the  statement  of  Newton's 
Second  Law.  Note,  however,  that  it  holds  in  this  form  only  when 
the  force  is  expressed  in  the  units  defined  above,  i.  e.,  in  absolute 
units.  If  the  unit  of  acceleration  be  taken  as  1  cm.  per  sec.,  and 
the  unit  of  mass  as  the  mass  of  1  cc.  of  water  at  4°  C.  (i.  e.,  1 
gram)  the  absolute  unit  of  force  is  called  a  "dyne."  A  dyne  of 
force  is  then  defined  as  the  force  required  to  impart  to  1  gram  of 
mass  an  acceleration  of  1  centimeter  per  second. 

There  is  another  unit  of  force  in  common  use,  which  is  called 

the  gram  of  force.      It  must  not  be  confused  with  the  gram  of 

mass.     The  gram  of  force  is  defined  as  the  force  of  the 

Number  of  3  . 

dynes  in  a       earth's  attraction  upon  a  gram  of  mass.     Since  this 
force  is  able  to  impart  to  the  gram  of  mass  an  accelera- 
tion of  980  cm.  per  second,  it  is  evident  that  one  gram  of  force  is 
equivalent  to  980  dynes  of  force. 

Experiment 

Object.  To  prove  that/  oc  ma. 

The  apparatus  is  the  same  as  that  used  in  Ex.  I,  save  that  the 

cord  c  (Fig.  2)  is  attached  to  the  frame  and  fork,  carried  over  the 

pulley  f)  and  fastened  at  the  other  end  to  the  weights 

Method.  £       i  •   i     •         j.      ,    -.1  T/.   AT.  •    T.J.  • 

»  wz,  one  of  which  is  adjustable.  If  the  weights  m  just 
balance  the  frame  and  fork,  the  removal  of  any  weight  ml  from  m 
will  cause  the  frame  and  fork  to  descend  with  an  acceleration 
which  may  be  measured  as  in  Ex.  I.  The  moving  force  will  be  ml 
grams,  the  mass  moved  will  be  the  mass  of  frame  and  fork  plus 


18 


MECHANICS 


After  leveling  the  instrument  as  in  Ex.  I,  adjust  the  mass  m  until 
the  fork,  when  given  a  slight  downward  movement,  will  just  con- 
tinue to  move,  but  without  acceleration.  This  elimi- 
nates the  friction  of  the  pulley  and  guides.  Now 
remove  a  mass  mi  (say  400  gm.)  from  m  and  obtain  two  traces 
precisely  as  in  Ex.  I.  Repeat  when  m  has  been  diminished  by  a 
second  400  gm.  so  that  the  moving  force  is  now,  say,  800  gm. 
(=  m2).  Measure  the  accelerations  in  the  two  cases  according  to 
the  method  given  in  Ex.  I,  using,  however,  but  one  unit  of  time,! 
say  that  of  20  vibrations  of  the  fork.  The  mass  of  the  frame  and 
fork*  as  well  as  the  masses  m,  ml7  and  m*  are  to  be  determined 
by  weighing  upon  a  pair  of  trip  scales.  The  test  here  made  of  the 
Second  Law  consists  in  verifying  equation  (11). 


Record 

Weight  of  frame  and  fork  ( W)  = .      Balancing  weight  (m)  =  — 


1st 
spaces 

trace 

2d  trace 
spaces        al 

1st  trace                3d  trace 
spaces         tt2      spaces         0*1 

S2  
S3~  —  

\— 

\ 

I  

\  1  ; 

1 

1                             \ 

f 

1 

f 

\                            \ 

\                            \ 

I 

f 

\                            \ 

Si 

I 

I 

\                            \ 

f 

f 

\                            f 

I 

I 

1                             1 

f 

f 

\                            \ 

S« 

Means 
Final  m 

fi_mi 

.-  *f     • 

[W+^-m^Xa,                  ^     ^ofcrro 

ft          ™>2 

^W+(m-m  2)]Xa2 

*The  mass  which,  at  the  circumference  of  the  wheel,  would  have 
a  moment  of  inertia  (see  Ex.  X)  equal  to  that  of  the  wheel  should  theo-j 
retically  be  included  in  this  weight.  This  is  usually  a  negligible  quantity] 

f  Do  not  simply  indicate  the  division.     This  blank  is  for  the  result 
the  division. 


NEWTON'S  SECOND  LAW  19 

Problems 

1.  Over  a  weightless  and  frictionless  pulley  (see  Fig.  5)  are 
suspended   masses  of   200  gm.   each.     A  weight  of    100    gm.  is 
added  to  one  side.     Find  the  acceleration  thus  imparted  to  the 
weights. 

Take  g,  the  acceleration  of  free  fall,  as  980 ;  the  acting  force  is  then 
100  X  980  dynes. 

2.  To  the  ends  of  a  rope  passing  over  a  weightless  pulley  are 
attached  two  bodies  of  unequal  masses,  one  of  200  gm. ,  the  other 
of  something  more  than  200  gm.    The  acceleration  imparted  to  the 
masses  is  observed  to  be  245  cm.  per  second.     Find  (1)  the  exact 
mass  of  the  larger  body;    (2)  the  tension  in  the  string.     Express 
the  tension  both  in  dynes  and  grams. 

SUGGESTION  (1). — If  #=the  difference  between  the  two  masses, 
then  the  acting  force  is  xg  dynes  and  the  mass  moved  is  400  -f-  &• 

SUGGESTION  (2). — Since  all  freely  falling  bodies  have  the  acceleration 
g,  the  force  of  gravity  acting  upon  any  mass  of  m  grams  is  mg  dynes.  If, 
because  of  some  retarding  force  (e.g.  tension  in  a  string),  the  downward 
acceleration  is  not  g  but  some  smaller  quantity  a,  it  is  at  once  clear  from 
the  statement  of  the  Second  Law  that  the  value  of  the  upward  or  retard- 
ing force  is  m(g  —  a)  dynes.  Thus,  if  the  body  is  at  rest,  the  upward  force 
is  mg  dynes.  If  it  has  an  upward  acceleration  amounting  to  a  units  the 
upward  force  is  m(g  -f-  a)  dynes. 

3.  A  10  gm.  mass  is  moving  with  a  velocity  of  40  meters  per 
second.    A  force  of  2000  dynes  opposes  its  motion.    How  soon  will 
it  be  brought   to  rest? 

Find  a  from  Second  Law,  then  t  from  Ex.  I. 

4.  A  900  kgm.  projectile  struck  an  embankment  with  a  velocity 
of  400  meters  per  second.      It  penetrated  4  meters.     Find  the 
resistance  in  kgms.  which  the  embankment  opposed  to  its  motion. 

Find  a  from  Ex.  I. 

5.  A  50-lb.  mass  hangs  from  a  spring  balance  in  an  elevator. 
How  much  does  it  appear  to  weigh  at  the  instant  at  which  the 
elevator  begins  to  descend  with  an  acceleration  of  400  cm.  per 
second?     (1  kilo  =  2. 2  Ib.) 

See  problem  2,  suggestion  (2). 

6.  A  man  pushes   steadily  upon  a  car  weighing  1000  kilos. 
After  5  seconds  it  is  moving  with  a  velocity  of  .5  meters  per 
second.     Find  the  man's  force  in  kilos. 

Find  a  from  Ex.  I. 


20  MECHANICS 

7.  A  man  who  can  jump  three  feet  high  on  the  earth  could 
jump  how  high  on  the  moon?  The  mass  of  the  earth  is  80  times 
that  of  the  moon.  The  diameter  of  the  earth  is  3f  times  that  of 
the  moon. 

SUGGESTION. — The  mathematical  statement  of  Newton's  law  of  grav- 
itation is  foe  —£-,  in  which/  is  the  force  acting  between  any  two  bodies, 

ra  and  M  their  respective  masses  and  r  the  distance  between  their  centers 
of  gravity.  By  the  application  of  this  law  first  find  the  relative  values 
of  \  the  acceleration  of  a  body  at  the  surfaces  of  the  earth  and  moon 
respectively,  then  note  that  the  initial  velocity  produced  by  the  spring 
is  always  the  same. 


Ill 

COMPOSITION  AND   RESOLUTION   OF   FORCES 

Theory 

COMPOSITION  OF  FORCES — RESULTANT. — From  the  statement 
that  "change  of  motion  is  proportional  to,  and  in  the  direction  of 

the  impressed  force,"  it  follows,  by  implication  at 
r^condL°awf  least?  that  a  given  force  always  produces  the  same 

change  of  motion  in  the  direction  of  its  action,  whether 
the  body  upon  which  it  acts  is  at  rest  or  in  motion,  whether  it  is 
acted  upon  by  other  forces  at  the  same  time  or  not. 

Consider,  for  example,  a  very  short  interval  of  time,  say  one- 
millionth  of  a  second,  during  which  a  force  /15  acting  in  the 

direction  AB  (see  Fig.  6),  is  accelerating  a  body  at  A. 
I  force.  At  .the  end  of  the  interval  the  body  will  have  acquired 
Iratwn  a  vel°city  by  virtue  of  which,  during  the  second  which 

begins  at  the  close  of  this  interval,  it  will  move  uni- 
formly and  in  a  straight  line  (see  First  Law)  to  some  point  B.     If 
the  interval  of  time  be  taken  suffi- 
ciently   small    the     acceleration 
during  the  interval  may  always 
be   considered  constant;  and  if, 
in  all  cases   considered,  the   in- 
terval   chosen    be   of   the    same 
length,  then  even  for  a  variable 
force  the  acceleration  will  be  sim-  FIGURE  6 

ply  the  velocity  acquired  in  one 

of  these  intervals  divided  by  the  length  of  the  interval.  Since, 
then,  the  velocity  acquired  in  an  interval  is  always  proportional  to 
the  acceleration,  the  line  AB  may  be  taken  as  representing  the 
acceleration  a^  due  to  the  force/!. 

If,  instead  of  receiving  an  acceleration  represented  in  magnitude 
and  direction  by  a^  the  body  were  acted  upon  by  a  force /2  acting 
in  the  direction  AD,  it  would  receive,  in  the  interval  considered, 


22  MECHANICS 

a  velocity  which  would  carry  it  in  one  second  to  some  point  D. 
AD  would  then  represent  the  acceleration  due  to  fz. 

If,  now,  the  two  forces  /t  and  /8  act  simultaneously,  the 
Second  Law  asserts  (by  implication)  that  the  effect  of  each  force  is 
the  same  as  though  the  other  did  not  act ;  i.e.,  that  in  the  short 
interval  considered,  the  body  will  receive  a  velocity  which  will 
carry  it  at  one  and  the  same  time  a  distance  AB  in  the  direction 
of  (i.  e.,  parallel  to)  AB  and  a  distance  AD  in  the 
oftwoef&rces  direction  of  AD.  This  is  merely  saying  that  the  veloc- 
ity acquired  in  the  interval  will  carry  the  body  in  one 
second  to  C.  Further,  the  path  between  A  and  C  must  be  a 
straight  line,  because  a  body  can  move,  by  virtue  of  an  acquired 
velocity,  only  in  a  straight  line.  Hence  the  line  AC  represents 
the  joint,  or  resultant,  acceleration  due  to  the  joint  action  of  the 
two  forces  /x  and  /2. 

But   one  single  force  /3  can  be  found  which,  acting  in  the 

direction  AC,  would  produce  this  acceleration  «3.     Such  a  single 

force  is  called  the  resultant  of  the  two  given  forces. 

Resultant        Thus  the  resultant  of  any  number  of  forces  is  defined 

force  defined.  • 

as  that  single  force  whose  action  would  produce  the  same 
'"''change  of  motion"  as  is  produced  by  the  joint  action  of  the  several 
forces. 

It  is  evident  from  the  above,  since,  for  a  given  mass,  forces  are 
proportional  to  accelerations,  that  the  resultant  of  any  two  forces 

acting  upon  a  particle  is  represented 

Resultant  of      .        .6      r  .  1 

any  number    in   intensity   ana   direction   by   the 
diagonal    of  the  parallelogram   the 
sides  of  which  represent  the  two  forces.      By 
the  successive  application  of  this  rule  to  the 
case  of  the  simultaneous  action,  of  three  or  more 
forces,  such  as  «,  #,  c  (see  Fig.  7),  the  follow- 
ing general   rule  is  obtained:     The   resultant 
FIGURE  7  C^)  °f  any  num^er  °f  forces-  is  represented  in 

magnitude  and  direction  by  the  line  which  closes 
the  polygon  whose  sides  represent  the  several  forces  when  the  latter 
are  conceived  as  acting  successively. 

The  problem,  then,  of  finding  the  magnitude  and  direction 
of  the  resultant  of  two  forces  a  and  b  which  include  an  angle 
0  (see  Fig.  S)  resolves  itself  into  the  trigonometrical  problem 


NEWTON'S  SECOND  LAW 


23 


"of    finding    the    length   of 

the  line  .c  and  the  value  of 

the    angle    9   in 

Calculation  „     .  , 

of  resultant     terms   o  i   the 

of  two  forces. 

given  magni- 
tudes  a,  b,  and  <£. 

To  find  the  length  of  c 
in  terms  of  a,  5,  and  <£,  drop 
a  perpendicular  pr  upon  a. 


Then 
Hence 
Now 
Hence 

But 


FIGURE  8 

<?  =  or  +  pr  .    But     pr  = 
c2  =  or  —rs  +  b2. 
or  =  (a  —  rs)   =  az  —  %ars+rs. 
cz  =  az  +  bz  —  %a  rs. 


Hence 


-r-  =  cos  \lt  =  —  cos 
o 

c*  =  «2  +  #2  + 


i.  e. 


cos 


Again  to  find  the  angle  0  in  terms  of  a,  b,  and 


—  =  sin  0  and  •—-  =  sin  \b  =  sin 
c  o 


Hence 


sn 


sin 


, 

or  sin  6  =  —  sin 
c 


(12) 


(13) 


The  resultant  of  three  or  more  forces  may  be  found  by  com- 
bining the  resultant  of  the  first  and  second  with  the  third,  this 
resultant  with  the  fourth,  etc.  A  better  method,  however,  is  given 
in  a  following  paragraph. 

RESOLUTION  OF  FORCES — COMPONENT. — The  component  of  a 
force  in  any  specified  direction  is  defined  as  that  force  which,  act- 
ing in  the  direction  specified,  would  produce  the  same 
effect,  so  far  as  motion  in  this  direction  is  concerned,  as 
is  produced  by  the  action  of  the  given  force.     Thus  if 
N  A  C  (Fig.  9)  represents  the  distance  over  which  a  body  would  move  in 
one  second  by  virtue  of   a   velocity  acquired 
through  the  action  of  the  given  force  during 
one  of  the  infinitesimal  intervals  above  consid- 
ered,   AB  evidently   represents    the    distance 
moved  over  toward  the  east  during  this  second ; 
FIGURE  9  i.e.,  if  A  C  represents  the  actual  acceleration, 


24  MECHANICS 

AB  represents  the  eastward  acceleration.  Or,  finally,  a  force 
represented  by  AB  is  equivalent,  so  far  as  motion  toward  the  east 
is  concerned,  to  the  given  force  A  C.  AB  is  therefore  the  com- 
ponent of  A  C  in  the  easterly  direction. 

Thus,  if  the  body  which  is  acted  upon  by  the  force  A  C  is  free 
to  move  only  in  the  direction  AB  (conceive,  for  example,  of  a  car 
on  overhead  rails  pulled  forward  with  the  aid  of  a  rope  by  a  man 
on  the  ground),  its  motion  under  the  action  of  the  force  A  C  must 
be  precisely  the  same  as  though  the  force  AB  were  acting  instead 
of  AC. 

The  process  of  finding  the  component  of  a  given  force  in  any 
direction  consists,  then,  in  finding  the  projection  in  the  required 
•direction  of  the  line  which  represents  the  given  force. 
Thus  AM*  (Fi&-  10)  is  the  component  in  the  direction 
AE  of  the  force  AF.     But  AM  =  AF  cos  6.     Hence, 
in  general,  the  component  in  any  specified  direction  of  a  given  force 
is  the  product  of  the  given  force  into  the  cosine  of  the  angle  included 
between  the  specified  direction  and  the  direction  of  the  given  force. 
Just  as  AM  is  the  component  in  the  direction  AE  ot  the  force 
AF  (see  Fig.  10)  so  A  G  is  the  component  of  AF  in  the  direction 

AH.     Now,  if   AE  and   AH  are   at 

ty  resolution       right  angles  to  each  other,   then   the 

figure   AGFM  is    a   parallelogram   of 
which  A Fis  the  diagonal  and  AM  and  AG  the 
sides.     Hence,  (see  above  under  RESULTANT)  the 
single  force  AF  might  entirely  replace  two  forces 
represented  by    AM  and  AG.      Conversely   the 
single  force  AF  must  be  in  every  respect  replace- 
FIGUBE  10         a^e  by  the  two  forces  A  M  and  A  G.      This  process 
of  replacing  a  given  force  by  its  components  in 
any  two  directions  at  right  angles  to  each  other  is  called  the  resolu- 
tion of  force. 

With  the  aid  of  the  resolution  of  forces  into  their  components 
in  any  two  rectangular  directions,  the  resultant  of  any  number  of 
forces  may  be  easily  obtained.     Thus :    Replace  all  of 
the  forces  ^i,  F*  ^3»  etc.  (see  Fig.  11),  by  their  com- 
Ponents   along    any   two   rectangular   axes   X  and  Y. 
Find  the.  algebraic  sum  x,  of  all  the  x  components  x^ 
x3,  etc.     Similarly  let  Y  represent  the  algebraic  sum  of  all  of 


NEWTON'S  SECOND  LAW 


25 


the  y  components  y,,  ?/2,  y^  etc.  The  resultant  of  x  and  Y  (see 
Fig.  11)  will  then  be  the  resultant  of  the  forces  F^  Fz,  F^  etc. 
The  direction  0  and  the  intensity  R  of  this  resultant  may  be  obtained 
from  any  two  of  the  following  simple  relations : 

Y  Y/x  =  tan  6     Y/R  =  sin  6  \       ^ 

If  a  particle 
which  is  subjected 
to   the  action   of 
forces  is  at  rest,  the 
.jgi    total  effectiveness 
of  the  forces  in  pro- 
ducing motion  along  any  particular  line 
must  be  zero,  i.e. ,  the  algebraic  sum  of  the 
components   of   the  forces  in 
condition  of    each  particular  direction  must 

equilibrium      .  _  „  , ,  . . .  _  J . 

of  a  particle,  be  zero.     If  the  position  of  the 
particle  be  taken  as  the  origin 
of  a  system  of  rectangular  coordinates  this 
condition  is  evidently  fulfilled  when  the 
sum  of  the  x  components  is  zero  and  the 
sum  of  the  y  components  is  zero.     Sym- 
FIGURE  11  bolically  the  condition  of  equilibrium  of 

a  particle  is  therefore 

^x  =  0  and  %  =  0  (15) 

in  which  2  is  the  sign  of  summation. 


I 


Experiment 

To  verify  the  laws  of  composition  and  resolution  of 
force. 

The  apparatus  consists  of  a  horizontal  force  table  (see  Fig.  12) 
about  the  circumference  of  which  may  be  set  at  any  desired  angles 
the  four  pulleys  «,  £,  c,  fZ,  over  which  any  desired 
weights  may  be  hung.  A  pin  holds  the  intersection  of 
the  cords  in  position  at  the  center.  When  a  test  for  equilibrium 
is  to  be  made  the  pin  is  removed. 


Object. 


Method. 


26 


MECHANICS 


1.  Set  two  of  the  pul- 
leys at  the  angles  marked 

30     and 


FIGURE  12 


Resultant  of 

two  forces.        an(J     apply 

weights  of  100  gm.  and 
150  gm.  respectively,  re- 
membering not  to  over- 
look the  weights  of  the 
pans  themselves.  Then 
calculate,  with  the  aid  of 
equations  12  and  13,  the 
direction  and  magnitude 
of  the  resultant.  Set  a 
third  pulley  180°  from 
the  calculated  angle,  ap- 
ply the  calculated  weight 

and  remove  the  center-pin.  To  show  that  the  equilibrium  is  not 
due  to  friction,  displace  the  junction  of  the  cords  and  tap  the 
table.  The  junction  should  return  to  the  center. 

2.  Resolve  each  of   the  three   forces  into    its  x  and    y  corn- 
Condition  of    Ponents   and  verify   the   condition  of  equilibrium  ex- 

equilibrium.     pressed  in  (15). 

3.  Insert  the  center-pin,  set  three  of  the  pulleys  at  any  angles, 
Resultant  of    and   aPPty  any  weights.       Calculate   by   means   of    a 
many  forces,    resolution  of  all  of  the  forces  into  their  x  and  y  com- 
ponents the  direction  and  magnitude  of 

the  force  requisite  to  produce  equilibrium. 
Test  as  in  1. 

4.  Fasten  one  end  of  a  spring  balance 
to  the  hook  in  the  wall  at  o  (see  Fig.  13) 

and  the  other  end  to  the  ring 

at  the  extremity  of  the  stick 
force.  '  mHf  From  the  ring  hang 
weights,  and  adjust  the  other  end  of  the 
stick  until  it  is  perpendicular  to  the  wall 
(see  Fig.  13).  Calculate  the  force  in  the 
spring  balance  and  compare  with  the  ob- 
served value.  Do  not  overlook  the  weight 
of  the  stick  itself.  FIGURE  is 


Vertical 


SECOND  LAW 


27 


1.  Forces 

1st 
3d 
.  •.  Resultant 


Record 
Direction  Magnitude  3.  Forces 


x  Comp.    y  Comp. 


2.  Forces 
1st 
2d 
3d 
Sum 


x  Comp.    y  Comp. 


1st 

3d 

3d 

Sum 

.  \  Resultant  magnitude  = 

.-.  Resultant  direction     =. 

4.  Force  in  spring  bal.  cal'd 
Force  in  spring  bal.  obs'd 


Problems 

1.  Show  that  the  acceleration  of  a  body  sliding  without  friction 
down  an  inclined  plane  of  length  I  and  height  li  is  g  x  j- 

2.  Show  that  the  velocity  acquired  in  sliding  down  the  plane  is 
the  same  as  that  acquired  in  falling  through  the  vertical  height 


,  vz.,  v  = 

3.  By  dividing  the  arc  ab  (Fig.  14)  of  a  vertical 
circle  into  a  large  number  of  small  inclined  planes 
and  applying  the  result  of  Problem  2,  show  that  a 
body  sliding  without  friction  down  the  arc  ab  acquires 
the  same  velocity  as  though  it  fell  vertically  from 
b  to  c.. 

4.  A  150-lb.  man  standing  in  the  middle  of  a  tight  rope  60  ft. 
long  depresses  the  middle  5  feet  below  the  ends.     Find  the  addi- 
tional   tension   in  the  rope  which  is  caused  by  his  weight. 

See  Fig.  13  for  suggestion  as  to  method. 

5.  A  bridge  span  is  5  meters  high  and  20  long  (see  Fig.  15). 
Find   the  vertical    pressure    and 

the  horizontal  thrust  upon  each 
of  the  piers  per  ton  of  weight  at  o. 

6.  A   bullet  is   fired  from   a 
gun  with  a  velocity  of  200  meters 
per   second   at  an  angle  of   30° 
with  the  horizontal.     How  high 

will  it  rise?     If  the  gun  is  on  the  FIGURE  15 


28  MECHANICS 

surface  of  the  earth  at  what  distance  from  the  gun  would  the 
bullet  strike  the  earth  if  there  were  no  air  resistance? 

7.  If  the  gun  mentioned  in  6  were  on  the  top  of  the  Eiffel 
tower  how  far  from  the  base  would  the  bullet  strike  the  earth? 
(Height  of  tower  =  335  meters.) 

8.  Show  that  the  time  of  descent  down  all  chords  which  start 
from  the  top  of  a  vertical  circle  is  the  same. 


IV 

THE  PRINCIPLE  OF   WORK 
THE  LAW  OF  MOMENTS 

Theory 

SCHOLIUM  TO  THIRD  LAW.* — The  Third  Law  asserts  that  force 
is  dual  in  its  nature,  that  a  change  in  the  momentum  of  a  body 
never  takes  place  without  an  equal  and  opposite  change 
in  the  momentum  of  some  other  body  or  bodies.     The 
reasons  for  this  assertion  and  some  of  its  consequences 
are  discussed  in  Ex.  VI. 

An  additional  and  wholly  distinct  assertion  is  made  in  the 
scholium  to  the  Third  Law.  It  is  that  the  work  done  by  the  force 
which  produces  the  motion  is  always  equal  to  the  work 
done  against  the  resistance  to  the  motion.  The 
scholium  reads:  "If  the  action  (activity)  of  an  agent  be 
measured  by  its  amount  and  its  velocity  conjointly,  and 
if  similarly  the  reaction  (counter-activity)  of  the  resistance  be 
measured  by  the  velocities  of  its  several  parts  and  their  several 
amounts  conjointly,  whether  these  arise  from  friction,  molecular 
force,  weight,  or  acceleration,  action  and  reaction  in  all  combina- 
tions of  machines  will  be  equal  and  opposite."  In  other  words, 
the  acting  force  F  times  the  velocity  v  of  its  point  of  application 
equals  the  resisting  force  R  times  the  velocity  vf  of  its  point  of 
application.  Symbolically  Fv  =  Rv' . 

Now  the  "work"  W  of  a  force  is  defined  as  the  product  of  the 
force  acting  F  and  the  distance  s  through  which  its  point  of 
The  definition  application  moves  during  its  action.  Thus  the  equation 
ofwrii.  W  =  Fs  .  (16) 

is  merely  the  definition  of  the  quantity  which  is  called  work. 

*See  Encyclopedia  Britannica,  Article  on  "Mechanics." 

29 


30 


MECHANICS 


Since  velocity  is  merely  rate  at  which  space  is  traversed,  i.  e., 
space  per  second,  it  is  seen  that  the  assertion  of  the  scholium  is 
then  simply  that  in  all  machines,  during  any  given  time, 
the  work  of  the  acting  forces  is  equal  to  the  work  of  the 
rces.     Symbolically, 


The  "wor/c 
principle. 


Fs  =  Rs'. 


(17) 


This  equation  represents,  then,  the  briefest  possible  statement  of 
the  "work  principle,"  which  is  the  essence  of  the  scholium  to  the 
Third  Law.  The  principle  is  a  generalization  from  experiments 
upon  all  sorts  of  machines.  It  is,  however,  to  be  observed,  that  in 
all  experiments  in  which  there  is  acceleration  the  resistance  to 
acceleration  which  the  body  offers  because  of  its  inertia  must  be 
included  among  the  reacting  forces,  and  must  be  put  equal  and 
opposite  to  that  force  whose  single  action  would  produce  the 
observed  acceleration,  i.  e.,  equal  to  ma,  m  being  the  mass  of  the 
body,  and  a  its  actual  acceleration. 

THE    PRINCIPLE    OF   MOMENTS. — The  great  principle  above 
asserted  can  of  course  be  tested  only  for  particular  cases.     One  of 
the  simplest  of  these  is  that  of  a  rigid  but  weightless 
^pie^appued  lever>  ^ree  t°  rotate  about  a  fulcrum.     Fj,g.  16  repre- 
uveraiQht      sen^s  the  force  diagram  for  the  case  of  such  a  simple 
lever,     ^represents  a  force  which  is  producing  a  clock- 
wise rotation .      W  represents  a  force  which  is  resisting  this  rotation. 

P  is  the  upward  force  exerted  by 
the  fulcrum  against  the  lever. 
If  the  fulcrum  be  a  knife-edge", 
and  the  bar  inflexible,  there  are 
no  frictional  or  molecular  forces. 
(Molecular  force  is  such  as  is 
called  into  play  by  bending  a  bar, 
stretching  a  spring,  etc.)  If  the 
motion  be  made  so  slowly  that 
the  acceleration  may  be  consid- 
ered zero,  then  the  weight  W  is 
the  only  resistance,  and  the  work 
principle  gives  Fs  =  Ws'  (see  Fig. 
16).  The  force  P  does  not  appear  in  this  equation  because  its 


FIGURE  16 


THE    SCHOLIUM   TO    NEWTON'S   THIRD    LAW 


31 


point  of  application  does  not  move;  hence  the  work  of  this  force 

I 
,• 
I 


s       I 
is  zero.     Now,  from  Geometry,  —  =  —  ,•     Hence 


FI=  m. 


(18) 


Worn 
principle 
applied  to 
rotation  of 
body  acted 
upon  by  any 
number  of 
forces  in  a 
plane. 


This  equation  is  rigorously  true  only  when  the  acceleration  is  reduced 
to  zero;  that  is,  when  the  bar  is  at  rest  or  is  moving  uniformly.  It 
represents,  therefore,  the  condition  of  equilibrium  of  the  bar. 

Next  consider  any  rigid  body  which  is  pivoted  at  o  and  acted 
upon  by  the  forces  F,  }\\,  W2  and  P  (see  Fig.  17).  ^is  produc- 
ing clock- 
wise rota- 
tion, Wl 
and  Wz  are 
resisting 
this  rota- 
tion, and  P,  the  pres- 
sure of  the  pivot,  has 
no  influence  upon  the 
rotation,  since  its  point 
of  application  does  not 
move.  Since  the  direc- 
tion in  which  the  force 
F  acts  is  not  parallel 
to  the  direction  in 
which  its  point  of  ap- 
plication moves,  the 
force  which  is  effective 
in  producing  the  mo- 
tion along  s,  the  direc- 
tion in  which  this  point 
of  application  of  F 
must  move,  is  not  .Pbut  the  component  of  F  in  the  direction  of 
5,  viz.,  F  cos  0.  Similarly  the  effective  resistances,  i.  e.,  the 
resistances  in  the  directions  Si  and  s2  in  which  the  points  of  appli- 
cation of  Wl  and  TF2  must  move,  are  not  Wl  and  TF2,  but  Wl  cos  015 
and  Wz  cos  6Z.  The  work  principle  then  asserts,  provided  there 
be  no  acceleration,  that 

(F cos  0)  s=( Wl  cos  00  sl  +  ( Wz  cos  02)  sz.        (19) 


FIGURE  17 


32  MECHANICS 

But  from  similar  triangles,  siSii  s2  =  h :  h\ :  h2.     Hence  (19)  becomes 

(F  cos  6)  h  =  ( Wl  cos  ft)  h,  +  ( Wz  cos  0,)  h2.          (20) 
But  again,  if  perpendiculars  /,  ?1?  4  be  dropped  from  o  upon 

the  lines  of  direction   of   the   forces  (see  Fig.),  then   cos  0  =  ^-, 

/L 

cos  ft  =  ^- »  cos  02  =  Y'     Hence  (20)  becomes 
A!  /£2 

*7  =  TFA  4-  TF2/2 .  (21) 

This  is  therefore,  according  to  the  "work"  principle  (the  scholium 
to  the  Third  Law),  the  condition  of  no  acceleration  about  o;  i.e., 
it  is  the  condition  of  rest  or  uniform  motion  of  rotation  (equilib- 
rium) of  the  body. 

Now  the  lever  arm  of  a  force  is  arbitrarily  denned  as  the  per- 
pendicular distance  from  the  line  of  direction  of  the  force 
?feiSoera!rm    *°  ^ie  ax*s  °f  Dotation.     Thus  £,  ^  and  4,  are  by  definition 
the  lever  arms  of  the  forces  F,  }}\  and  JF2,  respectively. 
The  "moment"  of  any  force  about  any  axis  is  defined  as  the 
Definition       Product  of  the  force  and  its  lever  arm.     Thus  Fl  is  by 
of  moment,      definition  the  moment  of  the  force  F  about  o. 

Hence  the  condition  of  rotational  equilibrium  (21)  expressed  in 
words  is  "The  sum  of  the  moments  of  the  forces  producing  clockwise 

rotation  must  be  equal  to  the  sum  of  the  moments  pro- 
Condition  of  *  1 

rotational       ducing  counter-clockwise  rotation;  or,  if  clockwise  rota- 

equilibrium.      , .        ,  , .  ,  ,      ,       . 

tion  be  called  negative,  and  counter-clockwise  positive, 
the  algebraic  sum  of  the  moments  of  all  the  forces  acting  must  be 
zero.  Symbolically 

3*7  =  0.  (22) 

Although  equation  (22)  has  been  developed  only  with  reference 

to  moments  about  the  axis  passing  through  o,  nevertheless,   if  the 

pressure  P  be  numbered  among  the  acting:  forces  the 

Equation  22  . 

holds  for        equation  also  holds  tor  moments  taken  about  any  ideal 

all  axes.  .  _  .  ..  .    .      ,         . 

axis  conceived  as  passing  parallel  to  the  original  axis 
through  any  point  whatever  in  the  plane  of  the  forces.  Thus  if 
(see  Fig.  18)  the  equation 

Fl  +  W^  +  W&  +  P10  =  0  (in  which  10  =  0) 

be  true  for  a  body  pivoted  at  o,  then  it  is  also  true  that,  if  o'  be  any 
point  whatever  in  the  plane, 

PL  =  0. 


THE    SCHOLIUM    TO    NEWTON'S    THIRD    LAW 


33 


\*7 
\ 

I^CH"""! 
•''''l'  \ 

l~, 

\    . 
1 

I 

\ 
\ 
\ 
\ 
\ 
\ 
\ 

This    conclusion     be- 
comes   self-evident 

from  the  consideration 

that,  if  the  body  be  in 

equilibrium,   P  is  the 

e<quilibrant,of  the 

three  forces  F,  TFi,  and 

W2'y    i.   e.,   so  far    as 

any    effects    produced 

by  jF,  Wi  arid  TF2,  are 

concerned,  these  three 

forces  may  be  replaced 

by   one     single     force 

equal  and  opposite  to 

P  applied  at  o.     The 

moment,  then,  of  this 

resultant  force   about 

any  axis  whatever  must 

be  equal  and  opposite 

to  the  moment  of  P; 

hence  the  sum  of  the 

moments  of  the  forces 

which  it  replaces,  viz.,  F, 

to  the  moment  of  P. 

If  a  body  of  mass  M  be  conceived  as  made  up  of  equal  infini- 
tesimal particles  each  of  mass  m,  then  the  center  of  mass  of  the 

body   is   defined  as 
that     point    whose 
distance  from  each 
of  three  coordinate 
planes  is  the  mean  of  the  dis- 
tances of  all  of  the  particles 
from   each    of    these    planes. 
Thus  if  n  represent  the  total 
number  of  particles,  and  x^  x% 

xn  the  distances  of  these 

particles  from  the  YZ  plane, 

then,  from  the  above  definition,  the  center  of  mass  must  lie  in  the 

plane  whose  distance  xc  from  the  YZ  plane  is  such  that 


FIGURE  18 


and  JF2,  must  be  equal  and  opposite 


FIGURE  19 


34 


MECHANICS 


05. 


-»HET-  (23) 

?i  ?i        w/t .        M 

Similarly  it  must  lie  in  planes  whose  respective  distances  yc  and  zt 
from  the  XZ  and  XY  planes  are  such  that 

and  za  =  —. TT- 


The  one  point  which  can  lie  in  all  three  of  these  planes  is,  by  defini- 
tion, the  center  of  mass  of  the  body.  If  the  three  coordinate 
planes  be  so  chosen  as  to  pass  through  this  point,  xc  =  0,  yc  =  0, 
zc  =  0.  Hence  for  this  case 
=  0, 


FIGURE  20 


=  0,  2mz  =  0.  (24) 

Or,  if  r  be  a  general  symbol  rep- 
resenting the  distance  of  a  par- 
ticle of  the  body  from  any  plane 
whatever,  the  center  of  mass  of 
the  body  is  that  point  which  is  so 
situated  that  a  plane  may  be 
passed  through  it  in  any  direction 
whatever  (see  Fig.  20),  and  yet 
the  following  condition  be  always 
satisfied : 

0.  (25) 

Of  course  distances  to  one  side  of  the  plane  are  reckoned  as  posi- 
tive, to  the  other  side  as  negative. 

If  a  body  be  placed  in  a  uniform  field  of  force,  *  equal  forces 
will  act  upon  all  the  equal  elements  of  mass.  Hence  the  point 
which  satisfies  the  condition  5/w  =  0  must  also  satisfy  the  condition 
3/r  =0  (see  Fig.  21). 
Now  the  center  of 
gravity  of 

Definition  ,      -,       . 

of  center  of  a  body  IS 
gravity. 

defined  as 

that  point  which  is 
so  situated  that, what- 
ever the  position  of 
the  body,  a  plane 

*  A  uniform  field  is  defined  as  one  in  which  the  force  acting  upon  a 
particle  does  not  change  as  the  particle  moves  about  in  the  field. 


FIGURE  21 


THE    SCHOLIUM   TO    NEWTON'S   THIRD    LAW 


35 


which  passes  through  this  point  and  is  parallel  to  the  direction  of 
the  force  always  satisfies  the  condition 

S/r  =  0.  (26) 

It  is  evident  that  if  a  body  has  any  center  of  gravity  at  all  this 
point  must  always  coincide  with  its  center  of  mass;  and  yet  a 
body  possesses  a  center  of  gravity  only  when  it  lies  in  a  uniform 
field  of  force.  Its  center  of  mass,  however,  is  a  perfectly  definite 
point  which  depends  in  no  way  upon  the  location  of  the  body. 

Now  a  rigid  body  placed  in  a  uniform  field  must  move  without 

rotation,  for  every  element  tends  to  move  at  every  instant  with 

the  same  velocity.     But  the  body  would  also  move  with- 

Property  of  , .          ...  .        „  ...  .    ,         . , 

the  center  of    out  rotation   if    a   single   force   (of    any   intensity  or 
direction)  were  applied  to  its  center  of  inertia.     For 
let  $!,  <£2  etc.  (see  Fig.  22)  represent  the  resistances  to  accelera- 
tion due  to  the  inertias  of  the  masses  m^ 
?/?2»  etc.     Since  for  the  center  of  inertia 
^mr  =  0,  and  since  inertias   are  propor- 
tional to  masses,  it  follows  that  3<f>r  =  0 ; 
i.  e.,  the  moments  of  the  resistances  due 
to  inertia  are  always  balanced  about  the 
center  of  inertia.      Hence  a  single  force 
applied    at    the    center    of    inertia    can 
never  cause  rotation.     It  is  evident,  then, 

that  a  single  force  applied  at  the  center  of  inertia  will  cause  exactly 
the  same  motion  as  is  produced  by  the  uniform  field.  This  force 
must,  of  course,  be  equal  in  intensity  to  the  sum  of  the  forces  ex- 
erted by  the  field  upon  all  of  the  particles  of  the  body.  Hence,  so 
far  as  all  effects  produced  are  concerned,  it  must  be  possible  to  replace 
the  action  of  the  field  by  the  action  of  such  a  single  force.  It  is 
therefore  customary  to  treat  a  body  which  is  acted  upon  by  gravity  as 
though  it  were  under  the  action  of  but  one  force,  that  force  being  ap- 
plied at  its  center  of  gravity  and  being  equal,  in  intensity  to  its 
weight.  Thus  the  center  of  gravity  is  sometimes  defined  as  that  point 
at  which  the  weight  of  the  body  may  be  conceived  as  concentrated. 

Experiment 

To  find  the  center  of  gravity  of  a  body;    to  verify  the  prin- 
ciple of  moments ;    and  to  study  the  theory  of  weigh- 

Object.  J  J 

ing. 


FIGURE  22 


36 


MECHANICS 


The  apparatus  consists  of  a  meter  bar  furnished  with  three 
knife-edges  a,  £,  and  c  (see  Fig.  23).  a  and  c  are  attached  to  a 

metal  frame  through  which  the  beam  may  be  slipped 
balance*61  an(^  ^°  wni°n  ^  may  be  clamped  by  means  of  the 

screw  s.  c  is  adjustable  in  a  vertical  plane;  b  is  fixed 
to  the  bar.  The  beam  is  also  provided  with  notches  m  and  m' ',  and 


FIGURE  23 


with  pans  P  and  P'  which  are  supported  from  the  top  of  the  bar 
by  knife-edges  n  and  ri '.  A  small  hole  passes  vertically  through 
each  of  the  knife-edges  c,  n,  and  n',  so  as  to  permit  plumb  lines  to 
be  dropped  from  these  points. 

DIRECTIONS.— 1.  Set  the  knife-edge  c  about  1  cm.  above  a. 
Support  the  beam  from  c  and  adjust  the  bar,  by  slipping  it  through 
the  frame,  until  it  balances  in  a  horizontal  position. 
Then  hang  P  and  P'  from  the  bar,  well  out  toward  the 
ends,  and  slide  one  of  them  along  until  the  bar  is  again 
horizontal.  Eead  off  the  lever  arms  upon  the  graduated  bar  and 
write  down  the  equation  of  moments.  Now  add  100  gm.  to 
P,  and,  without  altering  the  position  of  P',  slip  P  toward  the  ful- 
crum until  a  balance  is  again  found.  Write  again  the  equation  of 
moments.  The  solution  of  the  two  equations  thus  obtained  will 
give  the  weights  P  and  P '.  Check  the  results  by  weighing  on  the 
trip  scales. 


To  find 
weights  of 
pans. 


THE    SCHOLIUM   TO 


THIRD    LAW 


37 


2.  Remove  the  pans,  support  the  beam  this  time  from  # ,  and 
adjust  the  movable  knife-edge  and  the  bar  until  the  whole  system 

is  in  neutral  equilibrium  about  a\  i.  e.,  until  the  sys- 

Toflndcen-  •*.•         • 

terof  gravity  tern  shows  no  tendency  to  move  out  of  any  position  in 

of  system.  .  .   .     .     .  * .  . 

which  it  is  placed.  This  amounts  to  bringing  the  cen- 
ter of  gravity  of  the  system  into  coincidence  with  a.  A  small  bit 
of  soft  wax,  which  may  be  attached  to  the  bar  at  any  desired  point, 
greatly  facilitates  this  adjustment. 

3.  (a)  Having  brought  the  center  of  gravity  of  the  system  into 
coincidence   with    «, 

support  the  beam  from 
c,  hang  the  pans  from 


FIGURE  24 


uf  beam.  a  n  ^     place 

such  weights  upon 
them  that  the  beam 
assumes  a  position  in- 
clined about  45°  to  the 
horizontal  (Fig.  24). 
The  equation  of  mo- 
ments now  involves 
the  three  forces  TPi, 
JF2,  and  the  weight  of 
the  beam  PF3,  and  the 
three  corresponding 

lever  arms  ?x,  78,  13.  These  latter  are  to  be  measured  by  bringing 
a  half  meter  rule,  which  rests  horizontally  upon  some  firm  sup- 
port, up  to  the  plane  of  the  plumb  lines  which  hang  from  the 
knife-edges.  The  equation  of  moments  contains  but  one  unknown 

quantity,  viz.,  the  weight  of  the 
beam.  Solve  for  this  unknown 
quantity. 

(b)  Check  the  weight  thus  ob- 
tained by  supporting  the  bar  upon 
the  knife-edge  b  and  producing 
a  balance  by  means  of  a  known 
weight  W*  (see  Fig.  25). 

*  The  weights  of  the  pans  and  plumb  bobs  should  be  included  in  W,, 
W.,  and  W:  as  they  occur. 


FIGURE  25 


38  MECHANICS 

(c)  As  a  final  check  weigh  the  beam  upon  the  trip  scales. 

4.  (a)  When  the  knife-edges   n,  ri ',  and  c  are  not  in  the  same 
straight  line.     The  " sensitiveness"  of  a  balance  is  defined  as  the 

displacement   produced  when  some  arbitrarily  chosen 

Sensitiveness  %.          .    ,       .        ,  -,     ,  T.  J 

for  different  small  weight  is  added  to  one  pan.  it  should  decrease 
as  the  load  upon  the  pans  increases,  provided  the  sup- 
porting knife-edge  c  is  above  the  line  connecting  the  pan  knife- 
edges  n  and  n'  (see  Problems  2  and  3,  pp.  39,  40).  Set  the  knife-edge 
c  three  or  four  centimeters  above  the  line  connecting  n  and  n'. 
Support  the  system  from  c.  Hang  the  pans  from  the  beam,  but 
not  in  the  notches  (a  little  friction  at  the  knife-edges  n  and  n' 
vitiates  completely  the  results  of  this  experiment),  and  bring  the 
beam  into  the  horizontal  position.  Set  up  a  vertical  meter  stick 
behind  one  end  of  the  beam  and  read  off  upon  it  the  exact  deflec- 
tion produced  by  adding  very  carefully  one  gram  to  one  of  the 
pans.  Then  add  300  gm.  to  each  pan.  If  the  equilibrium  is 
destroyed,  reestablish  it  by  sliding  along  the  pan  to  which  the 
small  weight  was  not  added,  and  again  take  the  sensitiveness. 
Repeat  both  observations  to  make  sure  that  the  results  can  be 
duplicated. 

(b)  When  the  knife-edges  n,  n',  and  c  are  in  the  same  straight 
line.  In  this  case  the  sensitiveness  should  be  independent  of  the 
load  (see  Problem  2,  p.  39).  Lower  the  knife  edge  c  until  it  is  in 
line  with  n  and  n'.  Observe  as  in  4  (a).  (The  bending  of  the 
beam  may  still  .cause**  slight  dependence  of  sensitiveness  upon  the 
load.) 

5.  Hang  unequal  pans  P  and  P'  in  the  notches  n  and  n' ',  and 
slip  the  bar  through  the  frame  until  a  balance  is  obtained.     The 

arms  of  the  balance  are  now  unequal,  the  pans  are  of 
arrrecfweiah-  une(lual  weight,  and  the  center  of  gravity  of  the  beam 
with  false  js  not  beneath  the  point  of  support.     Nevertheless,  if 
an  unknown  weight  w  be  placed  in  the  pan  P  and  the 

li  I* 


THE   SCHOLIUM   TO    NEWTON'S   THIRD   LAW  39 

beam  brought  back  to  its  original  position  by  the  addition  of  a  known 
weight  a  to  the  other  pan,  the  following  equation  must  hold: 
wk  =  alz  (see  Fig.  26).  Now  place  win  P'  and  balance  by  means 
of  a  known  weight  b  placed  in  P.  The  equation  which  now  holds 
I  is  wlz  =  bli.  The  solution  of  these  two  equations  for  w  gives 
w  =  */ab.  If  a  and  ~b  have  nearly  the  same  value,  it  is  sufficiently 

correct  to  write  w=  — —     Hence  all  the  errors  of  a  balance  are 
eliminated  by  a  double  weighing. 

Record 

1.  Reading  at  a  in  condition  of  balance —  — 

Read'gforP  = forP'  = forP+100  = .*.  P= P'  =  — 

By  trip  scales  P= P'  =  —  %  error  in  Ps  = in  P's  =  — 

(b)  W  = lever  arm  of  W  =  -   -  of   W3  =  —         .-.   W3  =  — 

(c)  By  trip  scales  W3  =  -             %  error  in  (a)  =  —           in  (b)  =  — 
4.  No  load,  (a)  Reading  before  adding  1  gm. after dif. 


With  load,  (a) 

"      "      (b)  "  "        "        "  "  " 

5.  a=  —  —  b=  —         .'.  w= .      By  scales  w—  —         %  error  =  — 

Problems 

1.  Explain  why  a  balance  beam  returns  to  a  horizontal  position 
when  displaced  therefrom. 

2.  Fig.  27  represents  the  case  in  which  the  three  knife-edges 
ft,  ri,  and  c  are  in  the  same  straight  line.     Show  from  this  figure 

that  if  in  the  hori- 
zontal position  the 

12      "—•-.»_  Nc;  It  Jt     center  of  gravity  g  of 

the  beam  be  directly 
beneath  c,  i.  e.>  if  Ph 
=  P72,  then  in  the  dis- 
placed position  due  to 
the  addition  of  a  small 

w  *^,     .     weight   p    to   P,    the 

FIGURE  27  new  equation  of  equi- 


40 


MECHANICS 


FIGURE  £8 


librium,  viz/,  (P  +  p}l\  =  P'lfz  +  Wl^  reduces  to  pl\  =  W13.  Hence 
show  that  the  sensitiveness,  which  is  the  displacement  of  the 
beam  produced  by  p  (a  quantity  of  which  Z3  may  be  taken  as  the 
measure),  is  independent  of  the  load,  i.  e.,  of  P  and  P  '  . 

3.  Show  from 
Fig.  28  that  when 
n,  ri  ',  and  c  are  not 
in  the  same  straight 
line,  then,  after  dis- 
placement, Pl\  is 
not  equal  to  P'l\, 
and  hence  that  the 
sensitiveness,  i.  e., 
?3,  depends  upon  the 
load,  decreasing 
with  the  load  if  c  be  above  the  line  nri  ',  increasing  with  the  load  if 
c  be  below  the  line  nri  . 

4.  Does  the  period  of  vibration  of  a  balance  vary  with  the  load? 
If  so,  how  and  why? 

5.  With  a  given  load  how  should  the  period  be  aifected  by  a 
diminution  in  the  distance  between  the  knife-edge  c  and  the  center 
of  gravity  g? 

6.  A  circular  ring  weighing  5  Ib.  rests  hor- 
izontally upon  three  points  of  support  120°  apart. 
What  is  the  least  downward  force,  applied  to  the 
ring  in  a  direction  perpendicular  to  its  plane, 
which  will  cause  it  to  leave  one  of  the  points  of 
support?  voeeFig.  29.) 

7.  A  uniform  bar  weighing  10  Ib.  is 
pivoted  at  one  end  (see  Fig.  30).  A  hori- 
zontal force  of  5  Ib.  is  applied  to  the  free 
end.  When  a  condition  of  equilibrium  is 
reached,  what  angle  does  the  bar  make  with 
the  horizontal? 

8.  The  axle  of  a  wheel  carries  a  load  of  .  500 
kilos.     What  horizontal  force  must  be  applied  to 
the  axle  to  raise  the  wheel"  over  an  obstacle  12  cm. 
high,    the   radius   of    the   wheel   being   50  cm.? 
FIGURE  31  (See  Fig.  31.) 


FIGURE  29 


FIGURE 


THE    SCHOLIUM   TO    NEWTON 5S   THIRD   LAW  41 

9.  A  uniform  board  3  feet  square  and  weighing  25 
Ib.  rests  on  a  block  at  a  (see  Fig.   32),  and  is  kept 
from  falling  by  a  horizontal  force  at  B.     Find  the 
force  at  B  and  the  vertical  and  horizontal  pressures 

upon  the  block  at  a.  FIGURE  ca 

10.  Two  men  carry  between  them  a  weight  of  50 

kilos,  supported  upon  a  uniform  bar  weighing  30  kilos.  Where 
must  the  load  be  placed  in  order  that  one  man  may  carry  twice  as 
much  as  the  other? 

Choose  some  convenient  point  as  axis  and  apply  the  equation  2FI  =  0. 

11.  A  man  wishes  to  overturn  a  cubical  block  which  weighs 
100  kilos  and  has  a  5  ft.  edge.     In  what  direction  and  with  what 
force  must  he  push  in  order  that  he  may  accomplish  his  object 
most  easily?     After  the  block  has  once  started  will  the  required 
force  increase  or  decrease?     Why? 


V 

ENERGY   AND  EFFICIENCY 

Theory 

In  the  preceding  experiment  work  was  defined  as  the  prod- 
uct of  the  force  acting  and  the  distance  through  which  it 
moves  the  point  to  which  it  is  applied.  Symbolically, 
W=Fs. 

The  energy  of  an  agent  is  defined  as  its  capacity  for  doing 
work.  Energy  and  work  must  of  course  be  measured  in  the 
same  units ;  yet  it  is  obvious  that  they  are  not  synonymous  terms, 
for  a  body  may  possess  energy  and  yet  never  apply  it  to  the  pro- 
duction of  work.  Work  is  done  only  when  energy  is  expended. 

Since  work  is  a  product  of  force  and  space,  the  work  unit  must  of 
course  involve  a  force  unit  and  a  space  unit.     The  absolute  centi- 
meter-gram-second  (c.  g.  s.)  unit  of  energy  or  of  work  is 
energy  and     the  dyne-centimeter,  also  called  the  erg.     An  erg  of 
work  is  done  when  a  dyne  of  force  moves  the  point  upon 
which  it  acts  through  a  distance  of  one  centimeter.     Other  work 
units  are  the  gram-centimeter,  kilogram-meter,  foot-pound,  etc., 
the  definitions  of  which  are  evident  from  their  names.     Since  a 
gram  of  force  is  980  dynes,  it  is  evident  that  a  gram-centimeter  is 
equal  to  980  ergs. 

It  was  shown  in  Ex.  IV  that  Newton's  interpretation  of  the 

Third  Law  as  given  in  the  scholium  is  equivalent  to  the  statement 

that  in  all  mechanical  operations  the  energy  expended 

ergy.  Resist-    by  the  agent  is  equivalent  to  the  work  done  against  the 

ance  to  motion     J  &  *       .  £ 

is  inertia  four  resistances,  inction,  molecular  force,  gravity, 
and  inertia.  If  the  first  three  resistances  are  absent 
the  only  effect  of  the  force  is  to  impart  velocity  to  the  body.  By 
virtue  of  this  velocity  the  body  itself  becomes  possessed  of  the 
capacity  for  doing  work,  for  it  can  now  move  itself  against  a  fric- 

*  Electrical  and  magnetic  forces  are  here  classed  as  molecular. 

42 


THE    SCHOLIUM    TO    NEWTON 'S    THIRD    LAW  43 

tional  resistance,  compress  a  spring,  raise  itself  against  gravity,  or 
by  impact  overcome  the  inertia  of  some  other  body.  This  energy 
which  a  body  possesses  because  of  the  motion  which  has  been  com- 
municated to  it  is  called  "kinetic  energy." 

That  the  kinetic  energy  imparted  to  a  body  by  the  action  of  a 

force  is  exactly  equal  to  the  work  done  upon  it,  and 

PJjCflSfai    that  this  is  equal,  in  the  absolute  system  of  units,  to 

one-half  the  mass  of  the  body  into  the  square  of  its 

velocity,  may  be  shown  as  follows: 

Let  a  body  of  mass  m  acquire  a  velocity  v  under  the  action  of 
a  constant  force  ^acting  for  a  time  t,  and  in  that  time  moving  the 
body  a  distance  s.  The  work  done  upon  the  body  is  then  by  defi- 
nition Fs.  Now  let  the  body  be  brought  to  rest  by  being  subjected 
to  the  action  of  an  oppositely  directed  constant  force  F'  which 
requires  a  time  t'  and  a  space  s'  in  order  to  destroy  the  velocity  v. 
The  ''kinetic  energy"  of  the  body,  i.  e.,  the  work  which  it  is 
capable  of  doing  because  of  its  velocity,  is -by  definition  F's'.  But 
since  every  force  is  measured  by  the  rate  of  change  of  momentum 
which  it  produces,  the  first  force  F  is  measured  by  the  rate  at 
which  it  imparts  momentum,  and  the  second  force  F'  by  the  rate 
at  which  it  destroys  momentum,  i.  e., 

F '=  ma  and  F'  =  ma' .  (27) 

Now,  since  .Fand  F'  are  both  constant  forces,  i.  e.,  forces  which 
produce  uniformly  accelerated  motion,  by  Ex.  I 

s  =  %atz  and  s'  =  ±a't'*.  (28) 

Therefore  from  (27)  and  (28) 

Fs  =  $ma*t*  and  F's'  =  %ma'*t'*.  (29) 

But  by  hypothesis  the  velocity  imparted  by  F  and  the  velocity 
destroyed  by  F'  are  one  and  the  same  velocity.  Hence 

v  =  at  and  v  =  a't'.  (30) 

It  follows  from  (29)  and  (30)  that 

Fs  =  iwv8  and  F's'  =  %mv*. 
Hence 

Fs  =  F's'  =  %mv\  (31) 

Q.  E.  D. 

If  F  and  F'  are  variable  forces  it  is  only  necessary  to  conceive 
them  as  made  up  each  of  the  same  number  of  very  small  elements, 


44  MECHANICS 

each  element  being  a  constant  force,  v  will  then  be  the  velocity 
gained  under  the  action  of  one  of  these  constant  elements  of  .Fand 
destroyed  under  the  action  of  the  corresponding  element  of  F'. 
Hence  the  above  demonstration  is  perfectly  general. 

Next  suppose  that  the  resistance  which  the  working  force 
experiences  is  gravity  alone,  as  when  a  body  is  taken  from  position 
Potential  a  (see  ^]'S'  ^)>  and  placed  upon  a  hook  in  position  b. 
sManceis6'  ^ne  wor^  done  upon  the  body  is  the  force  of  the  pull 
molecular  times  the  distance  ab.  The  pull  is  a  variable  force, 
force  alone,  being  a  little  greater  than  the  weight  of  the  body  when 
the  motion  is  starting,  and  a  little  less  when  it  is  stopping.  The 
kinetic  energy  which  is  imparted  to  the  body  during  the  first 
instants,  when  the  velocity  is  being  acquired,  is  all 
given  back  during  the  last  instants  when  the  velocity 
is  being  lost.  When  the  operation  is  taken  as  a 
whole  no  velocity  is  imparted;  hence  the  only  re- 
sistance is  gravity.  Then  by  Newton's  interpre- 
tation of  the  Third  Law  the  work  which  has  been 
done  upon  the  body  is  equal  to  the  work  done  against 
gravity,  viz.,  the  force  of  gravity  upon  the  body 
times  the  distance  ab.  But,  in  this  case,  as  in  the 
FIGURE  33  preceding  in  which  the  resistance  was  inertia  alone, 
the  work  may  all  be  regained  without  the  expendi- 
ture of  any  more  work  on  the  part  of  the  agent.  For,  in  returning 
to  position  a,  the  body  is  capable  of  lifting  through  the  height  ab 
any  other  body,  e.g.  body  2  (see  Fig.  33),  whose  weight  does  not 
exceed  its  own.  The  ability  to  do  work  which  the  body  possesses 
by  virtue  of  its  position  at  b  is  called  its  potential  energy. 

If  the  work  be  done  against  molecular  force  alone,  as  when  a 
perfectly  elastic  spring  is  compressed,  the  work  done  can  all  be 
regained  by  releasing  the  spring,  which,  when  compressed,  is  pos- 
sessed of  potential  energy.  Potential  energy  is  in  general  any 
energy  which  is  put  into  a  system  by  a  change  in  the  position  of  its 
parts.  Thus  when  the  resistance  is  gravity  or  molecular  force 
alone,  an  amount  of  potential  energy  equal  to  the  work  done  is 
stored  up;  when  the  resistance  is  inertia  alone,  kinetic  energy 
equal  in  amount  to  the  work  done  appears. 

But  when  the  resistance  is  friction,  the  work  done  by  the 
agent  can  not  be  regained.  In  Newton's  day  it  was  supposed  to 


THE    SCHOLIUM    TO    NEWTON'S   THIRD    LAW  45 

have  disappeared  altogether;    but  about  the  middle  of  the  nine- 

teenth century  it  was  proved  by  Joule  that  for  every 

Resistance  fe   erg  of  work  which  so  disappears  there  always  appears  a 

friction  alone.       6_  .    .  .,     . 

perfectly  definite  quantity  of  heat;  hence  it  is  now 
customary  to.  say  that  the  work  expended  has  been  transformed 
into  heat  energy. 

The  experiments  of  Joule,  whereby  the  principle  of  the  equiv- 
alence of  heat  and  work  was  established,  consisted  in  transforming 

heat  into  work  in  as  large  a  variety  of  ways  as  possible, 
principle  of  e'  "  ^  means  of  the  friction  of  different  sorts  of  sub- 


S^ances5  DJ  percussion,  by  compression,  by  the  generation 
of  electric  currents  the  energy  of  which  was  finally  dis- 
sipated in  heat,  etc.  When  the  experiments  were  so  arranged  that 
the  heat  generated  was  taken  up  by  a  given  quantity  of  water,  it 
was  observed  that  a  given  expenditure  of  mechanical  energy  always 
produced  the  same  rise  of  temperature  in  the  water.  These  experi- 
ments had  much  to  do  with  securing  general  acceptance  for  the 
principle  of  conservation  of  energy,  a  principle  blindly  grasped  at 
by  philosophers  from  the  earliest  times;  first  stated,  in  the 
scholium  to  the  Third  Law,  by  Newton  in  1687,  but  with  respect  to 
mechanical  operations  only  ;  first  asserted  as  a  principle  of  universal 
applicability  by  the  German  physician  J.  R.  Mayer  in  1845;  first 
generally  accepted  and  universally  recognized  as  the  most  funda- 
mental and  most  fruitful  principle,  in  all  physical  science,  after 
Joule,  by  his  series  of  experiments  extending  from  1843  to  1878, 
had  demonstrated  the  equivalence  of  heat  and  ivork.  The  present 
accepted  value  of  the  mechanical  equivalent  of  heat,  i.  e.,  the 
number  of  ergs  of  work  required  to  raise  1  gm.  of  water  at  15° 
C.  through  1°  C.  is 

4.19xl07. 

But  the  principle  of  conservation  of  energy  is  more  than  the 
assertion  of  the  equivalence  of  heat  and  work.     It  may  be  stated 
thus:     Every  physical  (or  chemical)  change  of  condition 
rf     uas  a  fixed  mechanical  equivalent,  i.  e.,  can  be  equated, 
un(^er  a^  circumstances,  to  one  and  the  same  amount  of 
mechanical  work.     In  other  words,  whenever  a  change 
takes  place  in  the  condition  of  a  body  because  of  the  expenditure 
upon  it  of  mechanical  energy  (kinetic  or  potential),  the  change  is 
equivalent  to  the  work  done,  in  the  sense  that  if  the  body  can  be 


46  MECHANICS 

brought  back  to  its  original  condition  the  whole  of  the  energy 
expended  may  be  regained  either  in  the  form  of  work  or  the  equiv- 
alent heat. 

Thus,  applying  the  principle  to  a  mechanical  problem,  it  asserts 
illustrations  at  once  that  the  kinetic  energy  of  a  moving  body  is 
of  principle.  equaj  ^0  foe  work  done  in  setting  it  in  motion. 

Applying  it  to  a  chemical  problem  it  asserts,  since  the  burning 
of  1  gram  of  carbon,  i.  e. ,  the  formation  of  carbon  dioxide  'from 
carbon  and  oxygen,  generates  enough  heat  to  raise  97,000  gm.  of 
water  through  1°  C.,  that,  if  it  were  possible  to  directly  pull  apart 
the  united  carbon  and  oxygen  atoms,  97,000  x  4.19  x  107  ergs  of 
work  would  be  required  to  secure  1  gram  of  carbon  from  this 
compound.  Applied  to  an  electrical  problem  the  principle  asserts 
that  if  it  requires  1000  kilogram-meters  of  work  per  second  to 
drive  a  dynamo,  then  the  work  which  this  dynamo  does  per  second 
in  the  motors  which  it  runs,  plus  the  mechanical  equivalent  of  the 
heat  developed  in  all  of  the  machines,  and  in  all  the  connecting 
wires  must  be  exactly  equal  to  1000  kilogram-meters. 

The  principle  is   perhaps  the  most  important  generalization 

which  has  ever  been  made.      It  is  merely  an  extension  of  the 

scholium  to  the  Third  Law,  and,  like  it,  rests  upon  uni- 

Basisfor  .  ' 

assertion  of  versal  experience  rather  than  upon  any  one  particular 
experiment.  The  only  kind  of  direct  test  of  which  it 
is  capable  is  of  the  kind  which  Joule  made,  and  consists  in 
mechanically  producing  a  given  change  of  condition  (e.  g.,  a  given 
rise  in  the  temperature  of  water)  in  as  large  a  variety  of  ways  as 
possible,  and  observing  whether  the  work  required  always  cornea 
out  the  same.  Of  course,  such  an  experiment  tests  the  law  only 
for  one  particular -kind  of  physical  change. 

In  all  mechanical  devices  the  work  which  the  machine  accom- 
plishes is  inevitably  less  than  the  work  which  is  put  into  it,  for 
the  reason  that  there  is  always  some  friction  and  hence 
a  part  of  the  applied  work  disappears  as  heat.     Effi- 
ciency is  defined  as  the  ratio  of  the  work  done  by  the  machine  in 
any  given  time  to  the  energy  expended  upon  it  in  the  same  time. 

Experiment 

To  determine  the  efficiency  cufve  (1)  of  a  system  of 
pulleys,  (2)  of  a  water  motor. 


THE    SCHOLIUM   TO    NEWTON'S   THIRD   LAW  47 

1.  First  weigh  the  movable  block  of  pulleys  and  each  of 

the  pans  (see  Fig  34).  Then  give  to  JF,  which  includes  /- 
the  weight  of  pan  and  movable  block,  succes- 
sive values  of  about  300,  600,  1000,  1400, 
1800  grams,  and  find  the  corresponding  values  which 
must  be  given  to  P,  including  pan,  in  order  to  produce 
unaccelerated  downward  motion  of  P.  Calculate  the 
efficiency  corresponding  to  each  case,  and  plot  a  curve 
with  efficiencies  as  ordinates  and  loads  as  abscissae. 
Since  the  efficiency  is  the  ratio  of  the  ivork  of  the  force 
P  and  the  work  of  the  force  TF,  a  determination  of 
efficiency  must  involve  a  determination  of  the  ratio  of 
the  distances  through  which  the  points  of  application  of  P 
and  W  move.  This  can  be  obtained  without  a  measure- 
ment, as  a  little  consideration  will  show,  from  the  number  yy 
of  strands  between  which  the  weight  of  W  is  divided.  FIGURE  34 

2.  In  order  to  determine  the  energy  expended  upon  a 

water  motor  in  any  time,  it  is  necessary  to  know,  first,  the  pres- 
sure p  under  which  the  water  issues  from  the  orifice  o 
expended  (see  Fig.  35) ;  second,  the  volume  of  water  V  which 
issues  during  this  time.  The  energy  expended  upon 
the  motor  is  then  p  V.  This  will  be  evident  from  the  following 
considerations: 

Proof  No.  1. — Suppose  a  column  of  water  of  cross-section  o 
be  issuing  from  the  orifice  with  a  velocity  v.  Since  "pressure" 
means  force  per  unit  area,  the  force  driving  the  water  forward 
is  po.  This  force  carries  the  water  forward  a  distance  v  in  one 
second ;  hence  the  work  done  per  second  by  the  force  is  pov,  and  if 
the  experiment  last  t  seconds  the  total  work  done  is  povt.  But 
ovt  =  V.  Therefore  the  total  energy  expended  upon  the  machine 
during  the  experiment  ispV.  Q.  E.  D. 

Proof  No.  2. — Suppose  the  pressure  p  to  be  due  to  a  column  of 
liquid  of  height  7^,  and  suppose  a  mass  of  M  grams  of  liquid  to 
have  issued  from  the  orifice  o.  In  order  to  restore  the  condi- 
tions existing  before  the  mass  M  had  passed  through  the  orifice, 
it  is  necessary  to  raisf^f  throug^Jie  height  li  and  to  return  it  to 
the  liquid  in  the  reservoir,  i.  e.^Rs  necessary  to  do  a  quantity 
of  work  Mil.  ^jThis  therefore  represents  the  energy  which  has 
been  expended  in  the  passage  of  the  mass  M  from  the  orifice.  But 


48 


MECHANICS 


if  d  be  the  density  of  the  liquid,  p  =  lid. 


^-    But 


M 

d 


^-=  F. 


Mh=pV. 


d 
Q.  E.  D. 


In  this  experiment  friction  is  applied  to  the  axle  by  turning 
down  the  thumbscrew  s  (see  Fig.  35)  of  a  Prony  brake  until  the 

tension  in  the  spring  S  is  just  balanced,  i.  e.,  until  the 
fcy  °motwne  lever  arm  cn  C=  r]  °f  tne  brake  rests  midway  between 

the  stops  ^,  f.  It  is  generally  impossible  to  eliminate 
completely  oscillation  of  the  lever  arm,  but  its  mean  position  can 
be  estimated  with  sufficient  accuracy.  It  is  evident  that  the  ten- 


FIGURE  35 

sion  F  in  the  spring  8  represents  the  constant  pull  which  the 
machine  exerts  at  a  distance  cn  from  the  axle.  If,  therefore,  cn 
were  the  radius  of  a  pulley  upon  which  a  cord  were  being  wound 
up,  the  constant  pull  which  the  cord  would  exert  upon  any  load 
which  it  were  moving  would  be  equal  to  F.  One  revolution  of  the 
water  wheel  would  cause  this  load  to  move  a  distance  %-n-r.  It  is 
evident,  then,  that  the  work  accomplished  by  the  machine  in  N 


THE    SCHOLIUM   TO    NEWTON  'S   THIRD    LAW  49 


revolutions  is  %-n-rNF.  In  order  to  determine  the  value  of 
lower  end  of  the  spring  is  detached  from  the  lever  arm,  after  the 
run  has  been  made,  and  known  weights  added  to  the  spring  until 
it  is  stretched  to  the  length  which  it  had  during  the  run. 

The  motor  is  attached  to  the  regular  water  supply  of  the  room, 
but  irregularities  in  the  pressure  of  the  latter  are  equalized  by 
constant  introducing  before  the  motor  a  large  air-tight  tank  T 
pressure  (200  liters),  the  entrance  and  exit  both  being  at  the 

bottom.  It  is  thus  the  air  in  the  upper  part  of  the 
tank,  compressed  by  the  water-works  pressure,  which  is  the  imme- 
diate source  of  the  pressure  applied  to  the  machine. 

The  pressure  under  which  the  water  issues  from  the  orifice  o  is 
obtained  from  a  reading  of  the  mercury  manometer  AB.  If  there 

were  no  water  in  either  arm,  this  pressure,  measured  in 
measurement  centimeters  of  mercury,would  evidently  be  the  difference 

between  the  mercury  levels  in  the  two  arms  of  the 
manometer.  This  could  be  reduced  to  grams  by  multiplying 
by  the  density  of  mercury.  However,  since  the  arm  A  of  the 
gauge  fills  with  water,  and  since  it  is  the  pressure  at  the  level  of 
o  which  is  sought,  the  pressure  indicated  by  the  mercury  height 
must  be  diminished  by  that  due  to  a  water  column  of  height  equal 
to  the  difference  between  the  level  of  o  and  the  mercury  level  in 
A  .  In  solving  for  efficiency,  it  is  of  course  essential  that  p  be 
expressed  in  the  same  units  as  F. 

Starting  with  cock  K'  closed,  partially  open  cock  K  and  allow 
a  considerable  pressure  to  be  produced  in  tank  T.  Then  slowly 

open  K'  till  the  difference  in  the  levels  of  the  mercury 

in  the  pressure  gauge  is,  e.  g.,  100  cm.  Then,  while  one 
observer  holds  the  gauge-reading  constant  by  continually  adjusting 
K,  let  another  adjust  the  screw  s  till  the  lever  arm  en  maintains  in 
the  mean  a  horizontal  position.  These  adjustments  made,  at  an 
accurately  observed  time  deflect  the  discharge-water,  by  means  of 
the  flexible  rubber  tube  <?,  from  tank  E  into  the  empty  tank  /), 
and  al  the  same  time  let  a  third  observer  begin  to  count  the  revo- 
lutions of  the  speed  counter  (7,  which  is  attached  to  the  axle  by 
means  of  a  flexible  rubber  connection.  When  tank  D  is  about 
two-thirds  full,  stop  the  run  at  an  accurately  observed  time  by 
closing  cock  K'  '  .  Determine  V  by  weighing  the  water  in  tank  D 
upon  the  platform  scales.  Determine  Floy  stretching  the  spring 


50  MECHANICS 

S  by  means  of  known  weights  to  the  length  which  it  had  during 
the  run.  If  both  levels  of  the  mercury  in  the  gauge  were  not  read 
during  the  experiment,  reestablish  the  pressure  and  read  the  lower 
level.  Measure  the  lever  arm  en  [=  r],  and  compute  the  efficiency. 
Next  vary  the  tension  in  the  spring  S  by  raising  or  lowering  the 
support  to  which  the  upper  end  of  this  spring  is  attached,  and 
again  determine  the  efficiency  for  this  new  "load,"  the  pressure 
being  kept  the  same  as  before.  In  this  way  make  five  different 
runs  with  loads  which  vary  between  zero  and  the  maximum  which 
the  machine  is  able  to  carry  without  stopping  altogether.  The 
speed  will  then  vary  between  "racing"  speed  and  the  slowest  pos- 
sible speed.  Plot  a  curve  with  speeds  as  abscissae  and  efficiencies 
as  ordinates,  and  thus  determine  the  speed  for  which,  for  the 
given  pressure,  the  machine  is  most  efficient.  If  the  time  of  each 
run  be  made  of  exactly  the  same  length  the  speeds  will  of  course 
be  proportional  to  the  total  numbers  of  revolutions  N.  In  any 
case,  if  T  represent  the  number  of  minutes  of  duration  of  the 

experiment  and  n  the  speed,  n  =  —     In  this   experiment   it  is 

recommended  that  one  observer  regulate  and  read  pressure,  that 
another  attend  entirely  to  the  adjustment  of  the  Prony  brake 
(it  may  need  to  be  continually  regulated  throughout  the  run),  and 
that  a  third  observe  the  time  and  the  number  of  revolutions  of 
the  speed-counter. 

Record 

1.  Wt.  of  movable  block  = of  W  pan  = of  P  pan  =  - 

1st  value  of  W  =  —  of  P  =  —  .  •.  efficiency  =  — 

3d  value  of  W  =  —  of  P  =  —  ,.-.  efficiency  =  — 

3d  value  of  W  =  —  of  P  =  —  .:  efficiency  =  — 

4th  value  of  W  =  —  ofP=—  .  •.  efficiency  =  — 


2.  Mean  Hg  read 'g  in  B in  A Water  oor'n. cm.    .-.p gm. 

Wt.  of  D of  D+ water-    -    \V F T N .-.  eff.  - 

Second  run  "        "  "    -     -  " '* "—          " 

Third  run  "        "  "    -     -" " "- 

Fourthrun  "        "  "    -     -" " "- 

Fifth  run  "        "      "    " " " " 


THE    SCHOLIUM   TO    NEWTON *S   THIRD    LAW  51 

Problems 

1.  If  the  friction  between  two  surfaces  be  proportional  tx  the 
pressure  existing  between  them,  and  if,  in  the  experiment  with  the 
pulleys,  the  friction  due  to  the  bending  of  the  cord  in  passing  over 
the  pulley  be  a  wholly  negligible  quantity,  how  ought  the  efficiency 
of  the  system  of  pulleys  to  vary  with  the  load? 

2.  Solve,  Problems  2  and   3,  page  27,  from   a  consideration 
of  the  potential  and  kinetic  energies  of  the  body  at  the  top  and 
bottom  of  the  plane  and  arc. 

3.  Taking  the  distance  of  the  sun  as  90,000,000  miles,  the 
density  of  the  earth  as  5.50,  its  radius  as  4000  miles,  find  the 
kinetic  energy  in  kgm. -(meters)2,  which  the  earth  possesses  by 
virtue  of  its  orbital  motion.     Find  how  many  kilograms  of  water 
it  could  raise  from  0°  C.  to  100°  C.  if  its  energy  were  suddenly  to 
be  transformed  into  heat   by  a  collision.     Take  1  kilometer  as 
equal  to  .62  miles. 

Carry  to  three  significant  figures  only  and  use  throughout  the  expo- 
nential notation,  thus  7.14  X  1021. 

4.  What  mean  force  has  been  applied  to  a  kilogram  weight  if  it 
has  been  raised  5  meters  and  at  the  same  time  given  an  upward 
velocity  of  2  meters  per  second? 

5.  A  bullot  enters  a  target  with  a  velocity  of  120  meters  per 
second  and  penetrates  10  cm.     What  velocity  should  it  have  to 
penetrate  18  cm.? 

6.  A  Joule  is  107  ergs.     A  Watt  is  an  "activity"  of  1  Joule 
per  second.     A  horse-power  is  an  "activity"  of  746  Watts.     Find 
the  horse-power  of  a  steam  pump  which  lifts  100,000  liters  of 
water  per  hour  from  a  well  30  meters  deep. 

7.  How  many  Joules  of  work  are  expended  by  a  200-lb.  man  in 
climbing  stairs  60  ft.  high?     How  much  heat  would  he  develop  if 
he  were  to  fall  to  the  ground,  i.  e.,  how  many  grams  of  water 
would  the  heat  generated  by  the  fall  raise  through  1°  C.     (1  inch 
=  2.54  cm.,  1  kilo.  =  2.2046  Ib.) 


VI 
THE   LAWS   OF   IMPACT 

Theory 

If  two  bodies  be  connected  by  a  stretched  or  a  compressed 

spring,  then  if  the  spring  be  released,  common  observation  teaches 

that  both  bodies  are  set  into  motion.     According  to  the  Second 

Law  the  forces  which  produce  these  two  motions  are 

Meaning  and  »      i 

proof  of         measured,  by  the  rates  01  change  01  momentum  of  the 

Third  Law.  ,-    v.  ,-,        mi  .    ^    T 

two  bodies.  Hence  the  Third  Law  in  asserting  the 
equality  of  the  two  forces  asserts  the  equality  at  every  instant  of 
these  two  rates  of  change  of  momentum.  Experiment  alone  can 
furnish  proof  of  the  correctness  of  the  assertion.  But  the  Third 
Law  is  much  more  than  the  assertion  of  this  equality  in  this  par- 
ticular case.  It  asserts  that  all  forces  are  essentially  like  those 
existing  in  stretched  or  compressed  springs;  in  short,  that  all 
motions  in  the  universe  arise  from  stresses,  and  that  whenever 
one  body  is  set  in  motion  some  other  body  always  receives  the  same 
quantity  of  motion  in  the  opposite  direction.  This  assertion  is 
a  generalization  from  a  large  number  of  experiments  upon  forces 
whose  effects  can  be  observed  and  measured.  Astronomical  obser- 
vations attest  the  correctness  of  the  law  for  gravitational  forces. 
Laboratory  experiments  must  be  relied  upon  to  furnish  evidence 
regarding  forces  arising  from  impact,  magnetization,  electrification, 
etc.  In  this  experiment  the  law  is  put  to  the  test  for  certain  cases 
of  impact. 

When  any  moving  mass  m^  (see  Fig.  36)  makes  impact  with 
any  stationary  mass  mz,  it  is  observed  that 
the  latter   is    set   into    motion 

The  conser- 

vationof        while  the  former  loses  part  of 

momentum.      .  ,      .  ,         . 

its  velocity.     Consider  any  in- 
stant of  the  impact  and  let  «2  be  the  accel- 
FIGURE  33  eration  which    the   force  /*  (see   Fig.)  is 

52 


NEWTON'S   THIRD    LAW  53 

imparting  at  that  instant  to  m2,  and  #t  the  negative  acceleration 
which  the  force  fi  is  imparting  to  m^     The  Second  Law  gives 

/!  =  w^,      and     /2  = 
The  assertion  of  the  Third  Law,  then,  is 

/i  =/*,  or  Wi«i 
But  if  the  rates  of  change  of  momentum  ot  m^  and  mz  are  at  every 
instant  equal,  it  follows  that  the  total  momentum  imparted  to  m^ 
during  the  whole  time  of  impact  is  equal  to  the  total  (opposite) 
momentum  imparted  to  w2,  or  symbolically,  if  Wj  represent  the 
velocity  of  m^  before  impact,  vl  its  velocity  after  impact,  and  v2 
the  velocity  imparted  to  w2, 

WI(MI-VI)  =m2v2; 
or  by  transposition 

m^  =  miVi  +  m2v2i  (32) 

an  equation  which  asserts  that  the  total  momentum  before  impact  is 
equal  to  the  total  momentum  after  impact. 

If  m2  has  an  initial  velocity  uz->  precisely  the  same  line  of 
reasoning  gives 

m^Ui  +  m2u2  =  mvVi  +  m2v2.  (33) 

Thus  the  Third  Law  asserts  that  momentum  is  conserved  in  all 
impacts,  be  it  betiveen  elastic  or  inelastic  bodies. 

While  thus  the  Third  Law  asserts  that  there  is  never  any  loss 

of   momentum   in  an   impact,   it  does   not  assert   that  there   is 

no  loss  in  kinetic  energy.     The  mechanical  energy  is 

Uons  after       always  less  after  impact  than  before,  and  in  the  case  of 

an  inelastic  impact  between  two  bodies  one  of  which  is 

at  rest  (u2  =  0)  the  loss  can  be  theoretically  calculated  from  a  knowl- 

edge of  the  masses  alone,     Thus,  for  this  case,  since  in  inelastic 

impact  the  bodies  remain  together  after  the  collision,  Vi  =  vz,  and 

the  general  equation  (33)  becomes 

m^Ui  =  (raj  +  m2)  vz.  (34) 

Substituting  the  value  of  v.z  obtained  from  (34)  in  the  expression 
which  represents  the  fractional  loss  /  of  kinetic  energy,  viz., 
_  K  E  before  —  K  E  after  _  \m^.if  —  |  (mt  4-  mz)v2z       , 
KE  before 


there  results,  after  reduction,  the  simple  formula, 

(36) 


MECHANICS 


The  Third  Law  therefore  leads  to  the  interesting  conclusion  that 
the  per  cent    loss  in    kinetic  energy,   when  one  inelastic   body 
impinges  upon  another  which  is  at  rest,  is  altogether  independent      • 
of  the  velocity  of  the  impinging  body. 


Experiment 

To  test  the  equality  of  momenta  before  and  after 
impact  for  the  case  of  inelastic  impact. 
It  is  not  easy  to  measure  directly  the  velocities  immediately 
before  and  after  an  impact.     But  if  the  velocity  of  the  impinging 
body  be  acquired  by  a  fall  from  a  known  height  7^,  and 

Method.  .       £  i     i_    j  j    -L          i      -j.      •         •   • 

if  the  struck  body  expend  its  velocity  in  rising  to  a 
known  height  7^2,  the  required  velocities  uv  and  vz  can  be  easily 
calculated  from  the  measured  heights  7^  and 

~        t 


Object. 


In  order  to  realize  these  condi- 
tions a  lead  ball  m^  rendered  per- 
fectly inelastic  by  means  of  a  piece 
of  soft  wax,  is  allowed  to  fall  down 
a  circular  arc  from  the  position  d 
to  the  position  d'  (see  Fig.  37).  In 
so  doing  it  acquires  the  same  veloc- 
ity as  though  it  had  fallen  through 
the  vertical  distance  db  (see  Ex. 
Ill,  Problem  3,  and  Ex.  V,  Prob- 
lem 2).  After  impact  the  cylin- 
der and  ball  move  together  up  the 


FIGURE  37a 


XEWTOX'S    THIRD    LAW 


55 


Let   o 
together. 

Calculation 
of  7t2  from 
index 
readings. 


e. ,  the  reading  cor- 


arc  M,  which  is  graduated  in  degrees,  to  a  point  which  is  regis- 
tered by  the  light  aluminum  index  1.  The  vertical  height  cor- 
responding to  this  movement  up  the  arc  is  calculated  from  the 
index  readings  as  follows : 

be   the   index   reading  when   ball   and   cylinder   hang 
The  center  of  gravity  of  the  system  formed  by  the  ball 
and  cylinder  together  must  then  be  at  some  point  c 
(see  ideal  diagram,  Fig.  38)  which  is  directly  under- 
neath the  point  of  support  A.     Let  0'  be  the  index 
reading  when  the  cylinder  hangs  alone,  i. 
responding   to  the  position 
of  the  system  at  the  instant 
of  impact.     At  this  instant 
the  center  of  gravity  of  the 
system  is  at  some  point  c', 
which  is  to  the  right  of  c  a 
distance  such  that  c  —  c'  =  o 
—  o'.     Now  if  the  force  of 
the  blow  were  zero,  as  soon 
as  the  ball  and  cylinder  were 
joined,  the  force  of  gravity^ 
alone  would  cause  the  sys-  c/ 
tern  to  move  forward,  for  c, 
not  c'',  is  the  natural  posi- 
tion of  its  center  of  gravity.     The  velocity  acquired  in  falling  from 
c  down  to  c  would  of  course  carry  the  center  of  gravity  of  the 
system  up  again  to  some  point  c"  such  that  c  -  c  =  c"  -  c.     It  is 
evident,  then,  that  only  the  movement  over  the  arc  c"c'"  [=  o"k} 
can  be  attributed  to  the  velocity  imparted  by  the  blow.     Hence  liz 
is  the  vertical  height  corresponding  to  this  portion  only  of  the  arc, 
i.  e.  (see  Fig.  38): 

li^c"s.  (37) 

and  uv  =  uA  —  vA.  (38) 


But  c"s  =  uv, 

uA  ,  vA  ,. 

is  ow  -  =  cos  a,  and =  cos  0. 

r  r 

Hence  uA  —vA  =r  (cos  a  -  cos  0) 

in  which  0  is  the  angle  whose  arc  is  co"  [=  o/ 


(39) 

(40) 
and  a  is  the  angle 


whose  arc  is  c"c  [=  cc  =  oo'].     Since  the  arc  is  graduated  in  degrees, 


56 


MECHANICS 


a  and  6  are  obtained  at  once  from  the  readings.  Finally,  then, 
from  (37),  (38),  and  (40), 

Ji2  =  r  (cos  a  —  cos  0).  (41) 

The  radius  r  should  be  measured  from  the  point  of  support  to 
the  common  center  of  gravity.  Cos  a  and  cos  6  may  be  obtained 
from  any  trigonometrical  table. 

The  momentum  equation  which  it  is  sought  to  verify  is  mlul 
=  (mj  -fw2)v2.     But,   since  u^  =  V^gli^  and  v* 


tionsto  be       this  equation  may  be  written  in  the  form 

verified.  _  _ 

mi  v%  =  (mi  +  m2)  \///8.  (42) 

By  a  similar  substitution  the  expression  for  the  loss  of  kinetic 

energy  (35)  may  be  written 

' 


Make  several  preliminary  trials,  adjusting,  if  need  be,  the  posi- 
tions of  the  clamp  R  (Fig.  37),  and  the  suspensions  of  the  ball  and 
cylinder  by  means  of  the  thumbscrews  t  (Fig.  37a), 
until,  when  the  ball  is  released  by  burning  the  thread 
/,  the  cylinder  moves  smoothly  up  the  arc  without  wabbling. 
This  done,  measure  the  height  from  the  top  of  the  table  to  the  top 
of  the  ball  by  means  of  a  meter  stick  furnished  with  a  sliding  clip 
j  (Fig.  37).  Then  move  the  index  up  the  arc  to  nearly  the  point 
which  will  be  reached  by  the  cylinder.  After  the  impact  take 
very  carefully  the  index  readings  at  &,  o,  and  0',  and  also  the  dis- 
tance from  the  table  or  floor  to  the  top  of  the  ball  when  the  latter 
is  held  at  the  point  of  impact.  (It  must  of  course  be  provided 
that  the  reference  plane  from  which  the  heights  ab  and  ad  are 
measured  is  accurately  horizontal.) 

Obtain  at  least  two  sets  of  readings.  Weigh  the  ball  and 
cylinder  upon  the  trip  scales. 


Record 


1st  Trial 


ad  (Fig.  37) ab  -    -   .  •.  hl  - 

Read'g  at  k r  at  o  -  -  at  o>  — 

Mom.  bef. aft. %  dif .  - 

I  from  (43)  —  from  (36)  — 


2d  Trial 


ad    =- ab    = 

atk  = at  o  = 

r==—  0  =  - 

Mom.  bef. aft.  — 

I  from  (43)  = from 


r*i 

at  o' 


NEWTON'S  THIRD  LAW  57 

Problems 

1.  What  relation  exists  between  ml  and  mz  when  -J  of  the  K  E 
is  lost  in  heat?     When  ±?     When  f? 

2.  The  moon  moves  toward  the  earth  a  distance  of  15  ft.  per 
minute.     Find  how  far  the  earth  moves  toward  the  moon  in  the 
same  time.     (Ratio  of  masses,  1  to  81.4.) 

3.  A  rifle  bullet  weighing  20  gm.  was  fired  into  a  ballistic  pen- 
dulum weighing  4  kilos.     The  latter  moved  up  an  arc  a  distance 

corresponding  to  a  vertical  rise  of  5  cm.     Find  the  velocity  of  the 
bullet. 

4.  Is  it  true  that,  at  the  start,  the  wagon  pulls  back  with  the 
same  force  with  which  the  horse  pulls  forward?     If  so,  how  is  any 
motion  produced?     If  not,  reconcile  your  answer  with  the  Third 
Law. 

5.  A  bullet  weighing  20  gm.  and  having  a  speed  of  300  meters 
per  sec.,  struck  and  imbedded  itself  in  a  bird  weighing  5  kilos, 
which  was  flying  in  the  same  direction  as  the  bullet  with  a  speed 
of  150  kilometers  per  hour.     Find  the  velocity  of  the  bird,  in 
kilometers  per  hour,  the  instant  after  it  was  shot. 

6.  What  becomes  of  the  momentum  of  a  meteorite  which  col- 
lides with  the  earth?     What  becomes  of  its  energy? 

7.  Two  equal  inelastic  balls  moving  with  equal  velocities  in 
opposite  directions  collide.     Show  that  in  this  case  the  momentum 
Before  impact  is  the  same  as  that  after  impact. 

Note  that  velocity  is  a  directed  quantity. 

8.  A  billiard  ball  weighing  100  gm.  and  moving  east  with  a 
speed  of  2  meters  per  sec.  was  struck  by  a  putty  ball  weighing  4 
gm.  and  moving  south  with  a  speed  of  20  meters  per  sec.     Find 
the  speed  and  direction  of  the  ball  the  instant  after  the  impact. 

9.  A  500  gm.  bird  sat  on  a   pole  30    meters   high.      A  boy 
standing  20  meters  from  the  base  of  the  pole  shot  the  bird  with  a 
10  gm.  bullet  which  had  a  speed  of  150  meters  per  sec.     How  far 
did  the  bird  rise  above  the  pole?     How  far  from  the  base  of  the 
pole  did  it  strike  the  ground? 

Assume  that  the  bullet  lodged  in  the  bird. 


VII 

ELASTIC    IMPACT.      COEFFICIENT    OF   RESTITUTION 

Theory 

Since  force  is  equal  to  rate  of  change  of  momentum  it  follows 
from  the  Third  Law  that  the  mean  force  acting  between  two 
impinging  bodies  is  the  total  change  in  the  momentum 
°^  either  divided  by  the  time  of  duration  of  the  impact. 
Since  this  time  can  not  in  general  be  determined,  it  is 
customary  to  confine  attention  to  the  total  change  in  momentum 
which  each  body  experiences  by  virtue  of  the  impact.  This 
quantity  is  called  the  * 'impulse"  of  the  force,  and  will  be  here- 
after represented  by  the  letter  R.  If,  then,  u±  and  vl  represent 
the  velocities  of  the  first  body  m,±  before  and  after  impact 
respectively,  uz  and  vz  the  velocities  of  the  second  body  m^ 
before  and  after  the  impact,  then  by  definition 

R  =  ml(ul-v1),  (44) 

or  (see  Third  Law), 

R  =  mz(vz  —  u^).  (45) 

In  the  case  of  elastic  impact,  it  is  convenient  to  divide  R  into 
two  parts  RI  and  Rz,  of  which  R±  represents  the  impulse  during 
the  compression,  and  R%  the  impulse  from  the  instant  of  greatest 
compression  to  the  instant  of  separation. 

In  impacts  between  inelastic  bodies   Rz  =  0.     If  the  colliding 
bodies  are  perfectly  elastic,  it  might  be  expected  that  Rz  would  be 
equal  to  R^     In  point  of  fact  this  is  never  the  case, 
^coefficient    ^ol  there  are  losses  due  to  internal  friction  even  with 
ti(mStHu        bodies  which  when  subjected  to  static  tests  show  per- 
fect  elasticity   (see  definition   of   perfect  elasticity  in 
Ex.  VIII).     However  Newton  proved  experimentally  that  for  any 

two  given  bodies  the  ratio  -y^  is  always  a  constant  so  long  as  the 

*+i 

impact  is  not  so  violent  as  to  produce   permanent  deformation. 

58 


THIRD  LAW  59 

This  ratio  is  called  the  coefficient  of  restitution ,  and  will  be  here- 
after represented  by  the  letter  e.     It  is  always  less  than  unity. 

This  coefficient  of  restitution  e    =  —    may  also  be  shown  to  be 


[•a 


the  velocity  of  recession,  v2  —  v^  of  the  two  colliding  bodies  divided 

by  their  velocity  of  approach,  ut  —  u2.     For  it  is  evident 

c  =  y,—  v     that  at  the  instant  of  greatest  deformation  ml  and  m2 

Ui~u*      have  a  common  velocity.     Call  this  velocity  S.     Then, 

from  the  above  definitions  of  Rl  and  R^ 

A  =  mi  (ui  -S)=m2  (S-  M,),  (46) 

R,  =  mi  (S  -  Vl)  =  m2  (v2  -  S).  (47) 
Division  of  (47)  by  (46)  gives 

R2  r      ,      B-V!      v*-S  /AQ. 

•J*-  (48) 


From  (48)  come  the  two  equations, 

'8-i\  =  e  (?/!-£),  (49) 

v2-S=e  (8-u2).  (50) 

Addition  of  (49)  and  (50)  gives 

v*-vl  =  e  (wi-w«).  (51) 

Q.  E.  D. 

In  the   special   case   of    impact  against  a  fixed  plane,  u2  =  0, 
v2  =  0,  and  v±  is  opposite  in  direction  to  ?/t  therefore 

*--2f-  (52) 

M, 

Newton's  law  as  to  the  constancy  of  e  may  therefore  be  very  easily 
tested  by  varying  the  velocity  of  approach  of  a  body  toward  a 
fixed  plane  and  measuring  the  corresponding  velocities  of  rebound. 
If  the  body  be  dropped  vertically  upon  the  fixed  plane,  the 
velocities  uv  and  Vi  can  easily  be  determined  from  the  heights  of 
fall  and  of  rebound. 

The  actual  loss  L  of  mechanical  energy  upon  impact  (not  the 
fractional  loss  /)  is  evidently 

Ims  of  J 

L  =  (i™i"i*  +  4»W)  ~  (faff  +  ^10?)  ;      (53) 


impact. 


60  MECHANICS 


or  from  (44)  and  (45) 

7?  72  7? 

£=Y  (iii  +  v^-—  (u2  +  v2)=  —  {(Ui-uJ-fa-vJ}.  (55) 

The  combination  of  (55)  with  (51)  gives 

L  =  ~(Ul-u,)(l-e).  (56) 

But  substitution  in  (51)  of  the  values  of  v2  and  Vi  obtained  from 
(44)  and  (45)  gives 

R=jnsn*_  (l  +  e)(u      u)f  (57) 

mi  +  mt 

Substitution  of  this  value  in  (56)    gives  for  the  loss  of  kinetic 
energy 

£-ta-^-,tf  J55L.  (58) 

The  fractional  loss  I  is  this  expression  divided  by  the  initial  energy. 
Hence  for  the  simple  case  in  which  ra2  is  at  rest,  i.e.  ,  for  which  u2  =  0 

,  (.59) 


m 


and  for  the  still  simpler  case  in  which  mz  is  not  only  at  rest  but 
is  also  infinitely  large,  i.e.,  the  case  of  impact  upon  a  fixed  plane, 

l=(l-ez).  (60) 

Equation  (59)  shows  that,  as  in  the  case  of  inelastic  impact, 
the  loss  in  kinetic  energy  is  independent  of  the  velocity  of  the 
striking  body  provided  the  struck  body  is  at  rest.  When  e  =  0 
(59)  reduces  to  (36). 

It  appears  from  (58)  that  the  sole  condition  of  no  loss  of 
kinetic  energy  in  an  impact  is  e  =  1.  The  fact  that  e  is  always 
somewhat  less  than  unity  means  then  that  in  all  impacts  there  is 
some  transformation  of  mechanical  into  heat  energy. 

It  is  evident  from  the  general  momentum  equation  (33),  viz., 

Velocities  m^  +  m*U*  =  mM  +  ™^> 

after  impact   and  from  the  equation  (51)  which  determines  e,  viz., 

«  =  ^TT'  (61) 


that   the  velocities   after   impact    can   always  be   found   if    the 
coefficient  of   restitution,  the   two  masses,  and   the  two   initial 


NEWTON'S  THIRD  LAW 


61 


velocities  are  known.     For  the  simple  case  in  which  e  =  1,  i.  e., 

for  the  case  of  so-called  "perfectly  elastic"*  impact,  the  solution 
of  these  two  equations  gives 

~m»)  fnt)\ 

,  (bxi) 

~m*)  *        /AQ\ 
-•  (63) 


Object. 


Directions. 


Experiment 

1.  To  test  Newton's  law  as  to  the  constancy  of  e;  2.  to  prove 
the  equality  of  momenta  before  and  after  impact  and 
to  find  the  coefficient  of  restitution  and  the  per  cent 

loss  of  mechanical  energy  in  the  impact 

of  two  steel  balls. 

1.  Drop  steel  and  glass  balls  from 

the  clamp  c  (see  Fig.  39)  through  the 
ring  0,  and  slip  the  hori- 
zontal rod  r  up  or  down  the 

vertical  support,  until,  in  the  rebound 

from  the  smooth  top  of  the  heavy  steel 

plate  Pf,  the  bottom  of  the  ball  just 

becomes  visible  above  o.     Make  obser- 
vations   for    at    least  three    different 

heights  of  fall  which  lie  between,  say, 

30   cm.    and   100   cm.     In   each  case 

make  the  measurement  from  the  steel 

plate  to  the  bottom  of  the  ball. 

Since  in  general  for  falling  bodies 

starring  from  rest  v  = 


(64) 


FIGURE  39 


2.  The  form  of  apparatus  used  in  2  (see  Fig.  40)  is  the  same  as  that 
used  in  Ex.  VI,  save  that  all  three  heights,  viz. ,  the  height  of  fall  of  ml 

*The  term  is  rather  unfortunate  since  "perfect  elasticity"  is  often 
used  in  a  somewhat  different  sense  (see  Ex.  VIII). 

f  A  slab  of  slate  or  any  smooth,  hard,  and  heavy  body  may  replace  the 
steel  plate. 


62 


MECHANICS 


Apparatus. 


FIGURE  4) 


with  the  hand  as 
it  swings  back 
after  the  impact, 
and  take  the 
readings  at  5  and 
c.  Note  the  zero 
reading  of  each 
ball,  i.  e.,  the 
reading  when 
each  ball  hangs 
alone^  and  finally 
take  the  reading 
of  index  1  when 
ml  is  at  the  point 


Directions. 


before  impact  and  the  heights  of  rise 
of  both  mi  and  m2  after  im- 
pact are   measured   upon  a 
graduated  arc. 

First  make  such  adjustment  of  the 
lengths  of  the  supporting  strings  that, 
when  the  balls  hang  freely, 
the  wire  frame  carried  upon 
the  bottom  of  mz  clears  index  1   (see 
Fig.  41),  but   catches  index  2,  while 
the  like  frame  on  mi  catches  index  1. 
Then  adjust  so  that  when  the  thread 
which  holds  mi  back  is  burned,  mz  is 
driven  straight  up  the  arc  by  the  im- 
pact  from  mi.     Take  the  reading  of 
index  1  when  mi  is  tied  in  position  a. 
Next  slide  index  1  down  nearly  to  Z», 
the   point    to    which  mt 
will  move  after  the  im- 
pact (this  point  should  be 
approximately  located  by 
a  preliminary  trial),  and 
place    index    2    in     the 
neighborhood  of  c.    Then 
burn  the  thread,  catch  wi, 


FIGURE  41 


NEWTON'S  THIRD  LAW  63 

of  impact  d.  Tliis  is  not  tlie  reading  when  both  halls  hang  together, 
but  the  reading  when  m2  hangs  freely  and  m^  is  brought  down  so 
as  just  to  touch  mz. 

If  a  represent  the  difference  between  the  zero  reading  of  index  1 

and  the  reading  at  the  point  of  impact  d,  0  the  difference  between  the 

zero  of  1  and  the  reading  at  «,  /?  the  difference  between 

the  zero  of  1  and  the  reading  at  b,  and  to  the  difference 

between  the  zero  of  index  2  and  the  reading  at  c,  it  is  evident 

from  the  discussion  of  Ex.  VI  that  the  height  hi  through  which  w?.t 

falls  before  impact,  the  height  h9  through  which  mz  is  raised  by 

the  impact,  and  the  height  ///  through  which  m-i  rises  by  virtue  of 

the  velocity  which  it  retains  after  the  impact,  are  given  by  the 

relations  (see  Fig.  41) 

hi=  r  (cos  a  —  cos  0) ,    ) 
h*=  /-(l-cosco),  (65) 

Ui  =  r  (cos a- cos /?).   ) 
The  equation  to  be  verified,  viz., 

miUi  =  m^v*  +  m^Vi, 
may  be  written,  since  v  =  -v/2</A, 

m^lii  =  mz^lh  +  m^lii  ;  (66) 

or 

m^y  (cos  a  -  cos  0)  =  wz  */  (1  -  cos  CD)  -f  mi  \/(cos  a  -  cos  (3) .   (67) 
A  dmilar  substitution  in  the  expression  for  the  coefficient  of  resti- 
tution [see  (61)]  gives 

>/(!  -cos  w)  -  v/(cos  a  -  cos  )8) 

e  = = • 

^(COS  a  —  COS  6) 

The  loss  of  energy  is  to  be  obtained  from  e,  w^  and  mz  [see  (59)]. 


Record 


Steel 


Glass 
1st  2d    3d 


1.—  Ht.  of  fall          1st 3d 3d  - 

Ht.  of  rebound   "  -    -  "  - 

%  loss  of  K  E     "  -    -  "  -    -  "  - 

2. — Rdg.  of  index  1  at  a at  6 at  zero at  pt.  of  impact  — 

«      •<        "      2  "        at  c  

m^ m2 .'.  o  =  — -.'.  0  = .'.  /3  = .•.  w=r  — 

Momentum  before after e  = %  loss  of  K  E  =  — 


64:  MECHANICS 

Problems 

1.  Show  from  (57),  (44),  and  (45)  that  when  two  equal  balls, 
for  which  e  =  1,  collide  centrally,  they  simply  exchange  velocities, 
i.  e.,  the  result  is  the  same  as  though  one  had  passed  through  the 
other  without  in  any  way  influencing  it. 

2.  Hence  explain  why  the  only  effect  of  a  central  impact  of  one 
marble  upon  a  row  of  marbles  is  to  drive  off  the  end  marble.    Also 
why  the  impact  of  two  marbles  drives  off  two  from  the  end,  etc. 

3.  A  300  gm.  ball  approaches  a  bat  with  a  velocity  of  50  meters 
per  second;    it  leaves  with  an  opposite  velocity  of  100  meters. 
Find  the  mean  force  of  the  blow  if  the  impact  last  ^  second. 

4.  A  rapid-fire  gun  shoots   500   30-gm.   bullets   per   minute. 
Find  what  force  is  necessary  to  hold  it  in  place  if  the  velocity  of 
the  bullets  be  500  meters  per  second. 

For  a  constant  force,  or  for  a  succession  of  impulses  so  rapid  that  the 
effect  is  the  same  as  that  of  a  constant  force,  "rate  of  change  of  mo- 
mentum" is  change  of  momentum  per  second. 

5.  A  fire  engine  throws  400  liters  of  water  per  minute  from  a 
pipe  furnished  with  a  nozzle  of  4  cm.  diameter.     What  force  does 
a  wall  experience  against  which  the  jet  is  directed  at  short  range 
(assume  inelastic  impact)?     If  each  particle  of  water  were  "per- 
fectly elastic,"  i.  e.,  rebounded  with  the  velocity  of  approach,  what 
would  be  the  value  of  the  force? 

6.  A  bullet  weighing  50  gm.  is  fired  into  a  block  weighing  125  gm. 
Find  the  per  cent  loss  of  mechanical  energy.     Had  ball  and  block 
been  elastic  bodies  for  which  e  =1,  what  would  have  been  the  loss? 

7.  eh  very  nearly  unity  for  equal  spheres  made  of  perfectly 
elastic  materials  (steel,  glass,  ivory,  etc.),  but  it  is  not  unity  for 
unequal  balls  of  the  same  substances.     It  may  be  as  low  as  .  75  for 
steel  balls  of  greatly  different  size.     Thus  e  is  a  constant  of  the 
colliding  bodies,  not  of  the  material.     Why? 

Consider  vibration  losses,  and  the  conditions  under  which  vibrations 
will  persist  in  the  bodies  after  impact. 

8.  Find  the  velocities  after  impact  of  two  directly  impinging 
bodies  whose  masses  are  50  gm.  and  100   gm.,  whose  velocities 
before  impact  are  in  the  same  direction  and  have  values  of  600 
cm.  and  350  cm.  respectively,  and  for  which  e  is  unity.     Ditto  for 
balls  for  which  e  =  .90. 

9.  Explain  the  rise  of  a  rocket. 


VIII 

ELASTICITY 
HOOKE'S  LAW:     YOUNG'S  MODULUS 

Theory 

Most  substances  possess  in  greater  or  less  degree  two  quite  dis- 
tinct kinds  of  elasticity:  (1)  elasticity  of  volume,  (2)  elasticity  of 
form.  A  body  is  said  to  have  volume  elasticity  if  it 
^eil<^s  t°  return  to  its  original  volume  after  being  com- 
pressed or  dilated  by  the  application  of  force;  i.  e.,  if 
its  molecules  tend  to  maintain  fixed  distances  with  reference  to 
one  another,  and  resist  any  attempt  to  increase  or  decrease  these 
distances.  A  body  possesses  form  elasticity,  or  rigidity  if  its 
molecules  tend  to  maintain  a  fixed  configuration,  and  resist  any 
attempt  to  produce  slipping  motions  among  themselves. 

The  "volume  coefficient"  or  the  "volume  modulus"  of  elasticity 
is  a  constant  which  measures  the  restoring  force  called 
^°   P^ty  ^v  a  given    change   in   the   mean   distance 
between  adjoining  molecules,  the. configuration  remain- 
ing unchanged. 

The  coefficient  of  rigidity  or  "rigidity  modulus"  is  the  constant 
which  measures  the  restoring  force  called  out  by  a  given  change 
in  the  relative  positions  of  the  molecules,  the  mean  distance 
remaining  unchanged.  No  connection  whatever  exists  between 
the  two  kinds  of  elasticity.  Thus  all  liquids  possess  a  volume 
modulus  which  is  enormous,  a  rigidity  modulus  which  is  essentially 
zero.  India-rubber  has  nearly  the  same  volume  modulus  as 
water,  but  with  a  very  pronounced  rigidity  modulus.  The  metals 
have  very  large  moduluses  of  both  volume  and  form. 

A  body  is  said  to  be  perfectly  elastic  if  it  always  requires  the 
same  force  to  produce  the  same  displacement.     Thus  a  wire  would 
show  perfect  elasticity  if  the  successive  removal  of  a 
number   of    stretching   weights   caused    it   to   resume 
exactly  the  lengths  which  it  had  during  the  successive 
addition  of    the  weights,  and  that  no  matter  how  long  or  how 

G5 


66  MECHANICS 

short  a  time  the  weights  had  been  in  place.  If  static  experiments 
only  are  taken  as  the  test  of  perfect  elasticity,  all  liquids  are  per- 
fectly elastic,  and  most  solids  also  so  long  as  the  displacements  are 
kept  within  certain  limits.  The  limits  of  perfect  elasticity,  how- 
ever, differ  very  widely  for  different  substances.  Jellies  and  rubber 
show  perfect  rigidity  through  very  wide  limits,  iron  through  very 
small,  lead  through  smaller  still,  etc. 

There  is  no  connection  between  the  "degree"  of  elasticity  of 
a  body  and  its  elastic  constants.  The  former  measures  the  per- 

fectness  of  the  return  to  the  initial  condition,  the  latter 
dSKffs  ^e  magnitude  of  the  force  required  to  produce  a  given 

displacement.  Thus  jelly  is  nearly  perfectly  rigid,  but 
has  a  very  small  rigidity  coefficient.  Lead  has  large  coefficients, 
but  a  small  degree  of  elasticity.  In  popular  usage  a  body  is  said 
to  be  "very  elastic"  which  possesses  nearly  perfect  elasticity 
through  wide  limits.  Properly  speaking,  ability  to  rebound  is  a 
measure  of  resilience,  not  of  "degree  of  elasticity."  Neverthe- 
less, the  former  depends  in  large  measure  upon  the  latter.* 

Experiment  shows  that  within  the  limits  of  perfect  elasticity 
Hole's  ^e  displacement  produced  is  proportional  to  the  force 
Law.  applied^  whether  it  be  the  bulk  or  the  form  elasticity 

which  is  made  the  subject  of  test. 

For  the  determination  of  the  volume  modulus,  force  must  be 
applied  in  such  a  way  that  it  will  compress  or  dilate  the  body 

equally  in  all  directions,  but  will  not  distort  it.  This 
moduimme  can  °^  course  ^e  done  only  by  a  uniform  pressure  or 

tension  applied  to  all  points  of  the  surface  of  the  body. 
The  bulk  modulus  k  is  then  defined  as  the  ratio  between  the  force 
per  unit  area  (the  stress)  and  the  change  in  volume  per  unit  volume 
(the  strain).  By  Hooke's  Law  this  ratio  must  be  constant. 
Thus 

,          ,       /  force\      /change  in  vol.X 

vol.  mod.  =  I—    -)-(-        —, ), 

\  area  /      \         vol.          / 

or  symbolically,  since  by  definition  -    —  =  pressure, 


v  (69) 

V 


*Art.  on  "Elasticity"  by  Lord  Kelvin  in  Encyclopaedia  Britannica* 


COEFFICIENTS    OF    ELASTICITY 


67 


in  which  k  stands  for  the  volume  modulus,  P  for  pressure  (i.  e., 
force  per  unit  area) ,  V  for  volume,  and  v  for  change  in  volume. 
Unfortunately  the  direct  determination  of  Jc  is  not  easy.  It 
is  therefore  customary  to  determine  instead  a  quantity  called 
YOU  's  Young's  modulus,  which  is  a  cross  between  the  volume 
modulus.  modulus  and  the  rigidity  modulus.  A  known  weight  is 
hung  from  a  wire  and  the  elongation  measured.  Young"1 8  modulus 
is  then  defined  as  the  ratio  between  the  force  per  unit  cross-section 
of  the  wire  and  the  elongation  per  unit  length.  It  is  evident  that 
the  operation  here  described  tends  to  produce  a  change  in  form  as 
well  as  in  volume,  for  instead  of  increasing  or  decreasing  all 
dimensions  it  increases  the  length  and  decreases  the  diameter. 
Analysis  which  is  beyond  the  scope  of  this  book  shows  that  from 
this  hybrid  modulus  Y,  and  the  rigidity  modulus  n  (see  Ex.  IX) 
the  bulk  modulus  Tc  can  be  found  by  the  equation  .  ;i  't\ 


Y  = 


3k +  n 


(70) 


Experiment 


Object. 


To  test  Hooke's  Law  and  to  find  Young's  modulus  for 
steel. 

The  instrument  for  determining  the  elongation  produced  in  a 
wire  by  a  given  stretching  force  is  shown  in  Fig. 
42.     The   upper   end   of  the   wire   is 
firmly  clamped  in  a  chuck  at  a.     At  b 
the  wire  is  gripped  by  a  second  chuck  which  is 
set  into  a  cylindrical  brass  piece.     This  cylinder 
passes,  with  very  little  play,  through  a  circular 
hole  in  the  cross  piece  n.     n  is  rigidly  clamped 


Description. 


FIGURE  42 


68  MECHANICS 

to  the  upright  rods  R  and  7?',  and  carries  a  small  horizontal 
table  upon  which  rest  the  front  feet  of  the  optical  lever  m.  In 
order  to  prevent  easy  displacement  of  the  lever  these  feet  are 
set  in  a  groove  o.  The  rear  foot  of  the  lever  rests  upon  the  face 
of  the  chuck  b.  At  a  distance  of  twelve  or  thirteen  feet  from 
the  instrument  a  telescope  T,  to  which  is  attached  a  vertical  scale 
$,  is  so  placed  that  the  image  of  the  scale  formed  by  the  mirror 
m  is  visible  in  the  telescope.  The  addition  of  a  weight  to  P 
stretches  the  wire  and  lowers  the  rear  foot  of  the  lever  a  distance 
which  will  be  denoted  by  e.  If  I  represent  the  distance  from  the 
rear  foot  of  the  optical  lever  to  the  mid-point  between  its  front 
feet, it  is  evident  that  the  production  of  the  elongation  e  in  the 

wire  ab  has  caused  the  mirror  m  to  turn  through  an  angle  of  — 

radians.  This  angular  motion  of  the  mirror  causes  some  pointy 
instead  of  p  to  come  under  the  cross-hairs  of  the  telescope.  The 
beam  of  light  reflected  by  the  mirror  has  thus  been  turned  through 

the  angle  —  radians.     But  this  angle  is  twice  that  through  which 

the  mirror  turns  (see  Problem  1  below) ;  hen-ce  the  distance  c  is 
easily  determinable  in  terms  of  the  measurable  quantities^/,  mpt 
and  I. 

DIRECTIONS. — First  locate  the  image  of  the  scale  in  the  tele- 
scope.    To  do  this  move  the  head  about  near  the  telescope  until 
the  image  of  the  eye  is  seen  in  the  mirror  m.     If  the 

Finding  the  J,  _.   ,  .   . 

scale  in  the  eye  cannot  be  found  at  the  distance  of  the  telescope, 
move  the  head  up  toward  the  mirror  until  it  is  found ; 
then  move  back  again.  Keeping  the  image  of  the  eye  in  sight, 
move  the  telescope  and  scale  into,  or  near  to,  the  position  occu- 
pied by  the  eye.  Then  adjust  positions  until,  when  the  eye 
sights  over,  not  through,  the  telescope,  the  image  of  the  scale  is 
seen  in  the  mirror.  The  image  of  the  scale  must  then  fall 
upon  the  objective  of  the  telescope.  Then  looking  through 
the  telescope,  focus  by  means  of  the  rack  and  pinion  r  until 
the  mirror  itself  is  seen,  then  slowly  push  in  the  eye-piece  by 
means  of  r  until  the  scale  is  brought  into  view.  If  the  cross- 
hairs are  not  in  sharp  focus,  move  the  eye-piece  alone,  in  or  out, 
until  they  appear  perfectly  sharp,  then  refocus  upon  the  scale  by 
means  of  r. 


COEFFICIENTS    OF    ELASTICITY  69 

Next  turn  the  mirror  m  until  the  portion  of  the  scale  seen  in 
the  telescope  is  that  near  the  objective,  and  take  a  careful  reading 
of  the  position  of  the  cross-hairs  upon  the  scale,  esti- 
readings  mating  to  tenths  millimeters.  If  a  slight  change  in 
the  position  of  the  eye  changes  at  all  the  reading,  focus 
again  carefully  until  this  "error  of  parallax"  is  altogether 
removed.  Add  kgm.  weights  successively  to  the  pan  and  take  the 
corresponding  readings.  Make  similar  records  as  the  weights  are 
successively  removed.  If  upon  removal  of  the  weights  the  cross- 
hairs do  not  return  to  their  original  position,  the  limit  of  perfect 
elasticity  has  been  overstepped  and  the  readings  must  be  repeated 
with  the  use  of  fewer  weights.  In  adding  or  removing  weights, 
use  extreme  care  to  prevent  jarring  the  instrument  or  changing 
the  position  of  the  lever-point  on  chuck.  A  divergence  in  succes- 
sive elongations  of  more  than  one  per  cent  indicates  carelessness. 

In  order  to  determine  I  take  the  imprint  of  the  three  feet  of 

the  optical  lever  upon  a  sheet  of  paper.     With  a  knife-edge  draw  a 

line  connecting  the  centers  of  the  two  front  feet.     The 

mecmtre-        distance  from  the  middle  point  of  this  line  to  the  center 

of  the  third  foot  may  be  measured  with  a  steel  rule  held 

on  edge.     Estimate  to  tenths  millimeters.     Measure  the  distance 

mp  with  a  tape. 

Since  in  this  case  Young's  modulus  involves  the  sectional  area 
of  a  very  small  wire,  it  is  necessary  that  the  diameter  be  measured 
with  great  care.  Measure  with  micrometer  calipers,  and  let  the 
final  result  be  a  mean  of  at  least  a  dozen  observations  taken  at 
equal  intervals  from  top  to  bottom  of  the  wire.  In  using  the  cali- 
pers always  take  the  zero  reading  as  well  as  the  reading  when  the 
wire  is  between  the  jaws. 

In  the  calculation  of  Young's  modulus,  the  stress  (force  per 
unit  cross-section)  must  be  expressed  in  dynes  per  square  centi- 
meter, the  strain  (elongation  per  unit  length)  in  centi- 
uw  measure-   meters  per  centimeter.     It  is  also  required  to  plot  a 
curve  in  which  the  total  weights  in  the  pan  at  each  addi- 
tion shall  represent  abscissae,  and  the  corresponding  total  elonga- 
tions, measured  from  the  first  reading,  shall  represent  ordinates. 
As  a  check  upon  the  accuracy  of  the  work,  change  the  position 
of  the  telescope  and  scale,  and  make  a  second  complete  determi- 
nation of  Y. 


70 


MECHANICS 


Record 


1st  Determination 


2d  Determination 


Wts.  Edgs.         Difs. 


Diams. 


Rdgs. 


Difs. 


Diams. 


Means 
7  = mp  = wire  length  = 


,'.  F  = 


X  10 


F= 


X10 


Problems 

^  1.  From  the  optical  law  angle  of  incidence  equals  angle  of 
reflection,  prove  that  a  beam  of  light  reflected  by  a  mirror  turns 
through  twice  the  angle  through  which  the  mirror  turns. 

v0.  Can  a  body  whose  bulk  modulus  is  infinite  have  a  finite 
Young's  modulus? 

\t  3.  A  wire  80  cm.  long  and  .3  cm.  in  diameter  is  stretched  .3 
mm.  by  a  force  of  2  kilo.  How  much  force  would  be  required  to 
stretch  a  wire  of  180  cm.  length  and  8  mm.  diameter  through 
1  mm.? 

*  4.  An  iron  and  a  brass  wire  have  each  the  length  of  15  cm. 
when  each  is  stretched  by  a  force  of  1  kgm.  The  length  of  the 
iron  wire  becomes  15.4  cm.  under  a  stress  of  3  kgm.  and  that  of 
the  brass  wire  becomes  15.6  cm.  under  a  stress  of  9  kgm.  Com- 
pare the  modulus  of  iron  with  that  of  brass. 

5.  What  force  is  needed  to  double  the  length  of  a  steel  rod 
whose  diam.  is  2  mm.?     Assume  perfect  elasticity. 


IX 


Definition 
and 

illustration 
of  shear. 


THE   COEFFICIENT   OF   RIGIDITY 

Theory 

In  order  to  find  the  coefficient  of  rigidity  of  a  substance  it  is 
necessary  to  apply  a  force  which  will  cause  the  molecules  to  shift 
their  relative  positions  without  altering  at  all  their 
distances  apart.  Such  a  change  is  called  a  " shear." 
To  take  as  simple  a  case  as  possible,  imagine  a 
rigid  cylindrical  shell  whose  wall  is  one  molecule 
in  thickness,  and  let  the  height  of  the  cylinder  be  so  small 
that  it  contains  but  two  rows  of  molecules  [see  Fig.  43  (1)]. 
Let  a  tangential  force  act  upon  each  of  the  molecules  1,  2,  3,  4, 
etc.,  of  the  upper  row  so  as 
to  bring  them  into  the  posi- 
tions sjiown  in  Fig.  43  (2), 
the  molecules  of  the  lower 
row  being  held  fast  by  equal 
and  opposite  forces.  The 
change  produced  is  evi- 
dently a  pure  shear,  since 
configuration  alone  has  been 

changed,    all   distances   re-  FIGURE  43 

maining  exactly  as  at  first. 

The  shearing  force  is  the  total  force  which  has  acted,  or  the  sum 
of  the  forces  upon  the  individual  molecules.  The  angle  through 
which  the  line  connecting  any  two  molecules  which  originally  lay 
in  the  same  vertical  line  has  been  turned  by  the  shearing  force, 
is  taken  as  the  measure  of  the  shear.  This  angle  0  is  always 
expressed  in  radians. 

Were  the  cylinder  three  rows  of  moleculesjn  height  instead  of 
two  [see  Fig.  43  (3)],  then,  upon  the  application  of  the  same  shear- 
ing force,  the  upper  row  would  move  twice  as  far  as  before,  but 
the  angle  of  shear  6  would  remain  the  same,  for  the  case  would  be 

71 


MECHANICS 


precisely  the  same  as  though  the  middle  row  were  clamped  fast 
and  the  equal  and  opposite  forces  on  the  upper  and  lower  rows 

produced  each  the  same  effects  which  were  considered 
fnde  indent  *n  case  ^*  ^rom  an  extension  of  the  same  line  of  rea- 
of  cylinder  sonmg  t°  a  still  longer  cylinder,  it  is  evident  that  the 

shear  produced  by  the  application  of  a  given  shearing 
force  to  the  top  of  the  cylinder  is  independent  of  the  height  of  the 
cylinder,  for  the  shearing  motion  ultimately  ceases  only  when  the 
restoring  force  due  to  the  rigid  connection  between  the  top  row  and 
the  next  to  the  top  row  of  molecules  is  equal  to  the  shearing  force, 
i.  e.,  when  there  is  a  given  angular  displacement  between  these 
two  rows.  In  order  that  the  second  row  from  the  top  may  be  in 
equilibrium,  the  same  angular  displacement  must  exist  between  it 
and  the  third  row  as  exists  between  rows  1  and  2,  and  so  on  to 
the  bottom  row.  Thus  a  given  shearing  force  must  produce  a 
given  angular  tilt  in  a  row  of  vertical  molecules  whether  the 
cylinder  be  long  or  short. 

Conceive  now  the  ideal  cylindrical  shell  to  be  replaced  by  an 
actual  thin  hollow  cylinder  of  length  /,  of  mean  radius  r  and  of 

thickness  t  [see  Fig.  44  (1)].     Divide  the  upper  surface 

into  unit  areas,  and  let  a  tangential  force/  be  applied 

rigidity  n.        t()    each    unit   area  |-gee  ^  ^  ^j        ^   coejftcje}lt  Of 

rigidity  n  is  noiu  defined  as  the  ratio  between  the  force  per  unit  area 


FIGURE  41 


(the  stress)    and   the  shear  (the  strain)  produced    by    this  force. 
Symbolically  [see  Fig.  44  (1)], 


(71) 


COEFFICIENTS    OF    ELASTICITY  73 

The  total  shearing  force  is,  however,^/*.     Call  this  force  f  and 

shearing  force  n. 

the  area  01  the  ring  A  .     Then  n  =  -          —  *-  angular   dis- 

area 

placement;  or 


From  Hooke's  Law  this  ratio  is  the  same  for  all  values  of/'. 

It  is  not  so  easy  to  measure  0  directly  as  it  is  to  measure  <f> 

Measurement   \-F[8'  44    WL    the    an£le    through    which    the    end    of 

hollow0™  ^e  cylinder  is  twisted  by  the  force/'.  Since  the  arc 
cylinder.  a  [Fig.  44  (1)]  is  always  small  in  comparison  with 
the  length  of  the.  cylinder,  it  is  possible  to  write  without  apprecia- 
ble error, 

y  =  0.          But  also          ~  =  <fr. 

Hence  0  =  ^f-  (73) 

I 

Again  A  =  2«r*.  (74) 

Substitution  of  these  values  of  A  and  0  in  (72)  gives 

*-£»         .  .'?'•       (75) 

If  the  force/'  be  called  into  play  in  the  manner  shown  in  Fig. 
44  (3),  i.  e.,  by  clamping  one  end  of  the  cylinder,  screwing  the 
other  rigidly  to  a  grooved  circular  disk  and  applying  a  twisting 
force  of  say  F  [=  Wg~\  dynes  to  the  circumference  of  the  disk  by 
means  of  weights  TF,  then  by  the  principle  of  moments  [see  Fig. 
44  (3)], 


Hence  finally  from  (75) 


This  equation,  then,   expresses  w,  the  coefficient  of  rigidity,  in 
terms  of  quantities  all  of  which  are  easily  measurable. 

If,  as  is  usually  the  case,  it  is  found  more  convenient  to  twist 
a  solid  cylinder  rather  than  a  hollow  one,  formula  (76)  can  be 


74  MECHANICS 

transformed  to  fit  the  case  as  follows  :    Imagine  the  solid  cylinder 
to  be  made  up  of  a  large  number  of  concentric 
hollow  cylinders  of  equal  thickness  and 

Measurement 

of  n  from        of  radii  r^  i\,  rs  .  .  .  .  rn  (see  Fig.  45). 

solid  cylinder.  m         i  •   i 

The  moment  of  force,  say  Fh^  which 
it  is  necessary  to  apply   in   order   to   twist  any 
particular  hollow  cylinder  of  radius  rk  through 
FIGURE  45  the  angle  <£,  is  found  from  (76),  viz., 


.  (77) 

The  total  moment  of  force  Fli  which  must  be  applied  in  order  to 
twist  the  solid  cylinder  through  the  angle  <f>  is  evidently  the  sum 
of  all  the  moments  Fhk.  Thus  Fh  =  FH^  +  Fltz  +  FJi*  f  .  .  .  Fhn  = 


%irn<!>t  ,    .  o  ox       %Trn<t>  •        i  •   i    TI  •     .LI          T 

y  (r*  +  rf  +  rss  +  .  .  .  Tn)  =  —  -  —  x  —  in  which  R  is  the  radius 

of  the  solid  cylinder.*     Hence,  finally,  for  a  solid  cylinder, 

2  Fhl 


an  equation  which  shows  that  the  twist  <£  produced  by  a  given 
moment  of  force,  or  "  torque,"  Fli,  applied  to  one  end  of  a  cylin- 
drical wire  the  other  end  of  which  is  firmly  clamped,  is  directly 
proportional  to  the  length  and  inversely  proportional  to  the  fourth 
power  of  the  diameter  of  the  wire. 

It  is  evident  from  the  definition  of  the  coefficient  of  rigidity  n 

(also  called  the  "modulus  of  torsion")  that  it  is  a  constant  which 

is  characteristic  of  the  substance  and  is  independent  of 

^mmnSof     the  dimensions  of  the  particular  wire  used.     But  for  a 

torsion"  T  R 


E77, 

ticular wire,  the  ratio——  must  also  be  constant  (Hooke's  Law). 

This  constant  of  the  wire  is  technically  called  its  "moment  of 
torsion,"  and  is  represented  by  the  symbol  T0.  Thus  the  equa- 
tion 

*If  the  student  is  not  familiar  with  simple   integrations  he  may 
take  this  summation  for  granted.     Elementary  integral  calculus  gives 


C 
I 


- 
r*dr  =  —  ,  the  thickness  t  in  the  above  expression  being  the  same  as  dr. 


COEFFICIENTS   OF   ELASTICITY  75 


is  simply  the  definition  of  the  moment  of  torsion.  Expressed  in 
words,  the  moment  of  torsion  of  a  wire  is  the  constant  ratio  of  the 
moment  of  the  restoring  force  exerted  ~by  a  twisted  wire  and  the  angle 
of  twist.  If  in  (79)  <j>  =  unity  (i.  e.,  1  radian),  then  T0  =  Jli. 
Hence  the  moment  of  torsion  of  a  wire  is  sometimes  denned 
as  the  moment  of  force  required  to  twist  one  end  of  the  wire 
through  one  radian. 

It  is  evident  from  (78)  and  (79)  that  the  modulus  of  torsion  n 
may  be  expressed  in  terms  of  the  moment  of  torsion  T0  and  the 
dimensions  I  and  R.  Thus, 


Experiment 

(1)  To  test  Hooke's  Law  for  torsion;    (2)  to  determine  the 
moments  of  torsion  of  steel  wires  of  different  lengths 
and  diameters;  (3)  to  find  w,  the  coefficient  of  rigid- 
ity of  steel. 

Three    steel    wires  1,   2,    and   3  (Fig.    46)  are  provided,    of 

which  1  and  2  have  the  same  lengths  I  (about  1  m.)  but  different 

diameters,  1  and  3  the  same  diameters  (about  2.5  mm.) 

but  different  lengths.     By  means  of  a  set  screw  at  ^ 

the  wires  may  be  clamped  rigidly  to  the  grooved  and  graduated 

circular  wheel  C.     Displacements  produced  by  the  weights  W  are 

read  off  by  means  of  an  index  attached  to  the  frame.     Ball  bear- 

ings virtually  do  away  with  all  friction  and  render  possible  a  high 

degree  of  nicety  in  the  readings. 


FIGURE  46 


76  MECHANICS 

Clamp  wire  1  in  position  and  take  readings  first  as  100  gm. 
weights  are  successively  added  to  the  pan,  then  as  they  are 
removed.  Repeat  with  wires  2  and  3.  Measure  the 
Directions.  diameters  with  the  micrometer  calipers  (see  Appendix), 
taking  a  mean  of  a  large  number  of  readings  at  regular  intervals 
along  the  wire. 

Use  the  following  method  in  calculating  the  mean  twist  per 
100  gm. : 

If  the  total  twist  due  to  6  hundred  grams  is,  say,  4.32 

U       it  it  tt  tt          it    g  tt  .it  it       it        g    Q1 

tt  tt  ii  a  a  a  4.  a  a  a  tt  ^  §9 

a  a  tt  a  a  a  o  tt  tt  it  a  o  iiv 

tt  tt  tt  a  tt  "9  "  "  tt  tt  -\  A  A 

tt  it  tt  tt  tt  a  ^  a  tt  tt  n  iv  o 


"    "  sums  =  21  and  15.16 

then  the  most  correct  value  of  the  twist  per  100  gm.  which  can 
be  obtained  from  this  set  of  readings  is  15.16  •*•  21  =  .722. 

This  method  gives  to  each  observation  precisely  the  amount  of 
consideration  which  it  deserves.  Thus  it  gives  a  weight  of  6  to 
the  observed  displacement  for  600  gm.,  a  weight  of  2  to  the 
observed  displacement  for  200  gm.,  etc. 

Record 

Wire  No.  1  No.  2  No.  3  Diameters 


Wts.  Reads.  Difs?     Reads.  Difs.     Reads.  Difs.      No.  1    No.  2  No. 

0 

100 
200 
300 


0    -  ,          -  1 

200 

- 


400    -[_ 

500  -  -(_ 

600  •  -  f         -  f 

500    -  f  ~        -  f 

400    -  f  _ 

300 


f  ~        _  f 
I 

f 

/ 

100  f 


200 


0  r  -Means— 


COEFFICIENTS    OF    ELASTICITY  77 

I  of  No.  1  Mean  twist  of  No.  1 


_ 

I  of  No.  3  ~  '  Mean  twist  of  No.  3 

Diam.  of  No.  1  /Mean  twist  of  No. 


.  2\} 
.  I/ 


Diam.  of  No.  2  \Mean  twist  of  No. 

Radius  of  wheel  C .  •.  T0  for  1  —  for  2  —  for  8 

.  •.  n  from  1 from  2 from  3 Mean %  s  difs. 


Problems 

1.  If  one  of  the  wires  had  a  real  diameter  of  2.513  mm.,  but 
was  measured  as  2.501  mm.,  what  per  cent   of  error  was   thus 
introduced  into  n? 

2.  Decide  from  a  study  of  your  observations  which  one  of  the 
quantities  involved  in  n  introduces   the   largest   error   into  the 
result.     Why  is  it  needless  to  take  great  pains  in  measuring  the 
lengths? 

3.  Show  from  equation  (78)  that  n  would  be  correctly  denned 
as  the  moment  of  force  required  to  twist  a  cylinder  of  1  cm. 
length  and  1  sq.  cm.  cross-section  through  360°. 

4.  The  moment  of  torsion  of   a  particular  wire  is  7.21  x  106 
absolute  units;  its  diameter  is  2.732  mm. ;    its  length  is  50.1  cm. 
Find  the  moment  of  force  required  to  twist  a  wire  of  the  same 
material  of  1  mm.  diameter  and  4  cm.  length  through  90°. 

5.  A  man  grips  upon  the  circumference  of  a  bar  1,00  in.  long 
and  1  in.  in  diameter  and  twists  it  through  1°.     He  applies  the 
same  force  (not  .the  same  moment  of  force)  to  the  circumference 
of  a  bar  2  in.  in  diameter  and  80  in.  long.     Find  the  twist. 

*  Blanks  are  for  results  of  division. 


MOMENT    OF   INERTIA 

Theory 

It  was  experimentally  shown  in  Ex.  IV  that  the  condition  of 
rotational  equilibrium  of  a  rigid  body  acted  upon  by  the  two 
forces  F  and  W  (see  Fig.  16)  is  Fl  =  Wl' .  But  equilib- 
wtaS!mf  rium  is  reached  only  when  the  two  rates  of  rotation 
due  to  the  forces  F  and  W  are  equal  and  opposite.  It 
follows,  then,  from  Ex.  IV,  that  the  rate  at  which  a  force  can 
impart  angular  velocity  to  a  rigid  body  is  proportional  to  the 
product  of  the  force  and  its  lever  arm,  i.  e.,  to  the  applied 
moment  of  force.  Thus,  while  linear  acceleration  is  propor- 
tional to  the  acting  force ,  angular  acceleration  is  proportional  to 
the  acting  moment  of  force.  Hence  "moment  of  force"  bears 
precisely  the  same  relation  to  rotary  motion  which  force  bears 
to  linear  motion. 

The  inertia  of  a  body  is  that  property  by  virtue  of  which  it 

offers  resistance  to  acceleration.     The  measure  of  inertia  is  the 

resistance  offered  to  unit  acceleration;    or,  since  this 

resistance  is  always  equal  to  the  force  producing  the 

acceleration  (see  scholium),  the  measure  of  the  inertia  of  a  body 

is  the  force  necessary  to  impart  to  it  unit  acceleration.     This  was 

experimentally  proved  in  Ex.  II  to  be  proportional  to  mass ;    in 

the  absolute  system  of  units  equal  to  mass.     Hence  a  gram  of 

mass  has  one  unit  of  inertia,  two  grams  of  mass  two  units  of 

inertia,  etc. 

Moment  of  inertia  is  that  property  of  a  rotating  body  by  virtue 
of  which  it  offers  resistance  to  angular  acceleration.  It  is  meas- 
ured by  the  "moment  of  force"  necessary  to  impart  to 
inertiatof  '  ^ie  ^°^  un^  an9u^ar  acceleration.  Thus,  a  rotating 
body  has  unit  moment  of  inertia  if  it  requires  the 
application  of  a  unit  moment  of  force  (1  dyne-centimeter)  to 
increase  or  decrease  its  angular  velocity  at  the  rate  of  one  radian 
per  second;  it  has  two  units  of  moment  of  inertia  if  it  requires 
two  dyne- centimeters  to  impart  one  radian  of  acceleration,  10 
units  if  it  requires  5  dyne-centimeters  to  impart  an  acceleration 
of  i  radian  per  sec.,  etc.  Symbolically,  if  /represent  moment  of 

78 


MOMENT    OF    INERTIA  79 

inertia,  Fli  the  acting  moment  of  force,  and  a  the  angular  accel- 
eration produced, 


This  equation  is  to  be  regarded  merely  as  the  definition  of  /.  It  is 
thus  seen  that  "moment  of  inertia"  is  a  perfectly  definite  phys- 
ical quantity  which  can  be  determined  for  any  body  whatever  by 
merely  applying  a  known  moment  of  force  and  measuring  the 
angular  acceleration  produced.  The  definition  of  /  might  be  put 
in  the  following  form  :  Moment  of  inertia  is  that  physical  prop- 
erty in  which  two  rotating  bodies  agree  when  it  requires  the  same 
moment  of  force  to  give  to  each  a  given  angular  acceleration. 

It  is  evident  that  /  is  not  proportional  to  mass  alone,  as  is 

inertia,  for  everyday  experience  teaches  that  two  rotating  bodies 

mav  have  precisely  the  same  mass  and  vet  offer  widely 

Calculation       .*  ,  .  •   ,  ,-.  4.-  * 

of  moment      different  amounts   of   resistance   to   the  operation  of 

of  inertia.  .  ,       ,  »       •.  .   -, 

starting  or  stopping;  e.g.,  two  wheels,  one  of  which 
has  its  mass  concentrated  near  the  axle,  the  other  on  the  circum- 
ference. Thus  /  is  a  function  both  of  mass  and  of  the  distribution 
of  mass,  i.e.,  of  the  distances  of  the  elements  of  mass  from  the 
axis.  In  order  to  calculate  /,  the  moment  of  force  necessary  to 
impart  a  radian  of  acceleration  must 
be  found  in  terms  of  the  masses  of 
the  particles  and  their  distances  from 
the  axis.  Take  a  single  particle  mi  at 
a  distance  TI  [=0^]  (Fig.  47)  from  the 
axis  and  think  of  it  as  moving  inde- 
pendently of  all  the  other  particles  FIGURE  47 

under  the  action  of  a  force  /i  which 
gives  it  a  linear  acceleration  a{.     The  second  law  gives 
The  moment  of  the  force/!  about  the  axis  is/^v 
Hence  /i^  =  ???1a1r1. 

But  since  —  -  =  a  it  follows  that 


Similarly  the  moment  of  the  force  /2  which  is  necessary  to  give  to 
mz  the  angular  acceleration  a  about  the  axis  is 

/2r2  = 


.80  MECHANICS 

The  total  moment  of  force  Fli  which  must  be  applied  to  give  all 
the  particles  of  the  body  the  angular  acceleration  a  is  manifestly 
the  sum  of  the  moments  applied  to  the  several  particles.  Thus : 

Fh  =/!/*!  +/2r2  -f-  etc.  =  a  (m^*  +  m^r*?  +  etc.)  =  a2mr*          (82) 
or 

Fli 

—  =  lmr\  (83) 

i.  e.,  the  moment  of  force  necessary  to  impart  unit  angular  accel- 
eration is  Swr2.     But  this  is  by  definition  /  (see  81). 
Hence  /  =  2mr2.  (84) 

In  order,  then,  to  calculate  /  for  any  body,  it  is  necessary  to 
multiply  the  mass  of  each  particle  in  the  body  by  the  square  of  its 
distance  from  the  axis  of  rotation,  and  then  to  find  the  sum  of  all 
these  products.  For  an  irregular,  non-homogeneous  body,  this 
would  evidently  be  an  impossible  undertaking.  Hence  for  such 
bodies  the  moment  of  inertia  can  not  be  calculated.  It  can  only 
be  obtained  by  direct  experiment,  i.  e.,  by  applying  a  known 
moment  of  force  and  observing  the  angular  acceleration  produced 
(or  by  means  of  some  experiment  which  is  equivalent  to  this).  But 
for  certain  regular,  homogeneous  bodies,  it  is  possible  to  perform 
the  summation  indicated  and  hence  to  check  an  experimental 
value  of  /  by  means  of  a  calculated  one.  This  summation  is  done 
with  the  aid  of  the  integral  calculus.  Only  a  few  results  of  such 
summation  will  be  indicated  here. 

The  moment  of  inertia  of  a  uniform  cylinder  of  radius  R  and 
mass  M  rotating  about  its  own  axis  is 

(85) 


*  This  may  be  obtained  as  follows  :  If  <r  represent 
the  density  of  the  cylinder  and  I  its  length,  then  with 
the  use  of  polar  coordinates  (see  Fig.  48),  an  element 
m  of  mass  may  be  taken  as 

m  =  o-lr  dr  dd 

Hence 


FIGURE  48         I=2mr^  = 


=ffl  C* 

\}o 


The  other  integrations  are  somewhat  more  complicated  and  will  not  here 
be  attempted. 


MOMENT    OF    INERTIA  81    . 

The  moment  of  inertia  of  a  uniform  sphere  of  radius  R  and  mass 
M  rotating  about  an  axis  passing  through  its  center  is 

I=^MR\  (86) 

o 

The  moment  of  inertia  of  a  uniform  rectangular  bar  of  length  /, 
width  a,  thickness  £,  and  mass  M,  rotating  about  an  axis  parallel 
to  #,  and  passing  through  its  center  of  gravity  is 

/=S(?+a2)-  (87) 

From  a  comparison    of   the  equations /  =  ma  and  F/i  =  Ia  it 
appears  that  /  bears  precisely  the  same  relation  to  rotary  motion 
which  mass  bears  to  linear  motion.     /  is  therefore  some- 
*imes  called  the  mass  of  rotary  motion.     Whenever  m 
aPPears  in  an  expression  which  relates  to  linear  motion 
/  always  appears  in  the  corresponding  expression  which 
relates  to  rotary  motion.     Thus,  for  example,  the  kinetic  energy  of 
linear  motion  is  £rav2.     The  kinetic  energy  of  rotary  motion  is 
^/o>2,  to  being  the  angular  velocity  of  the  rotating  body.      This 
may  be  shown  as  follows :     The  kinetic  energy  Jce^  of  the  particle 
ml  (see  Fig.  47)  is  ^m-p*. 

But  Vi  =  0)7*!. 

Hence  lce\  =  \m^r^^. 

Now  the  total  kinetic  energy  KE  of  the  body  must  be  the  sum  of 
the  kinetic  energies  of  its  parts. 

Therefore  KE  =  2£rar2<o2  =  |/o>2.  (88) 

Q.  E.  D. 

Experiment^ 

To  determine  the  moment  of  inertia  of  a  circular  disk  by  apply- 
ing a  given  moment  of  force  and  measuring  the  corresponding 
angular  acceleration,  and  to  compare  the  result  with 

Object.  &x,  •  ' 

the  theoretical  value  01  1. 

A  disk  weighing  several  kilograms  is  mounted  upon  ball  bear- 
ings so  as  to  rotate  with  very  little  friction  about  its  own  axis  (see 
Fig.  49).     While  a  weight  m  imparts  to  the  disk  an 

Description.  *          '  . r  .  PI*. 

angular  acceleration  a,  an  electrically  driven  fork  of 
known  period  writes  a  trace  upon  the  blackened  face  of  the  disk. 
The  angular  acceleration  a  is  determined  precisely  as  were  the 
linear  accelerations  in  Exs.  I  and  II,  i.  e.,  by  subtracting  succes- 


82 


MECHANICS 


L  t 


s' 


sive  angular  distances  traversed  during  successive  equal  intervals 
of  time.  These  angular  intervals  are  obtained  in  degrees  from 
readings  made  upon  a  circular  scale  with  which  the  unblackened 

face  of  the  disk  is  provided.  By 
means  of  the  rack  and  pinion  s, 
the  frame  which  carries  the  fork 
may  be  moved  through  ways  in  a 
direction  parallel  to  the  face  of 
the  disk,  so  that  the  traces  made 
during  successive  revolutions  of 
the  disk  need  not  interfere.  The 
screw  sr  shifts  the  whole  disk  in 
the  direction  of  its  axis,  and  thus 
makes  it  easy  to  secure  a  suita- 
ble pressure  of  the  stylus  against 
the  blackened  face  of  the  disk. 

Wrap  a  fine 
thread  three,  or 
four  times 
around  the  cir- 
cumference, at- 
taching one  end 
to  the  disk  by 
means  of  a  small 
bit  of  wax.  To 

the  free  end  attach  just  enough  weight  to  equalize  the  friction 
of  the  ball  bearings  and  of  the  stylus  as  it  bears  upon  the  face 
of  the  disk.  Set  the  fork  in  vibration,  adjust  the 
stylus,  add  100  grams  to  the  thread,  and  then  suddenly 
release  the  disk,  at  the  same  time  moving  forward  the  fork 
by  means  of  s.  As  soon  as  m  touches  the  floor,  stop  the  disk, 
remove  it  from  the  frame,  and  carefully  mark  off  the  trace  as  in 
Ex.  I,  taking  a  group  of  50  waves  as  the  unit.  Eeplace  the 
disk  in  the  frame,  set  the  cross-hairs  of  the  low-power  microscope 
t  upon  the  limiting  mark  of  the  first  group  of  waves,  and  take  the 
reading  of  the  cross-hairs  of  the  microscope  t'  upon  the  circular 
scale  graduated  upon  the  unblackened  face  of  the  disk.  Then 
turn  the  disk  until  the  second  mark  comes  underneath  the  cross- 
hairs and  read  again.  From  such  readings  a  is  easily  obtained. 


FIGURE  49 


Directions. 


MOMENT    OF    INERTIA 


83 


It  must,  of  course,  be  expressed  in  radians  (see  definition  of 
moment  of  inertia).  Take  at  least  two  traces,  using  different 
masses  for  m\  e.  g.,  let  ml  =  100  gm.,  mz  =  200  gm. 

The  force  F  which  produces  the  rotation  of  the  disk  is  evi- 
dently the  tension  in  the  thread  q.  This  is  not  the  force  -acting 
upon  the  mass  m,  viz.,  mg,  for  a  part  of  the  acting 
force  mg  is  expended  in  producing  the  acceleration  a 
which  is  imparted  to  m.  By  the  scholium  to  the  Third 
Law,  mg  =  F+  ma,  or  F  =  m(g  —  a).  Since  the  acceleration  of  the 
weight  is  the  same  as  the  acceleration  at  the  circumference  of  the 
disk,  a  =  a  7?,  R  being  the  radius  of  the  disk  and  «  its  angular 
acceleration  in  radians.  Hence  the  acting  moment  of  force  Fh 
may  be  found  from  the  measurement  of  the  three  quantities  a, 
R,  and  m.  Fh  must  of  course  be  expressed  in  dyne-centimeters. 
From  Fh  and  a,  /  is  at  once  obtained  (see  definition  of  / ),  The 
theoretical  value  of  /,  see  (85),  involves  only  the  mass  M  and  the 
radius  R  of  the  disk.  M  is  to  be  obtained  by  weighing  upon  the 
platform  scales. 

Record 


1ST   DETERMINATION 

2D   DETERMINATION 

Angle         Angular               Angular 
Read'gs        Spaces                   Ace's 

| 

Angle         Angular               Angular 
Read'gs        Spaces                   Ace's 

•                             f 

I  ;  ( 

f                            J 

I                              ' 

i                         i 

f                            ' 

f                        i 

I                             f 

I                         f 

f                            1 

f 

/                             j 

[  — 

(                             / 

1                             | 

| 

Mean  ace  — 



J2                                 •   /  — 

Mean  ace.  —  — 

.     T  —                       Mpan    T  — 

I 
Problems 

1.  A  constant  pull  of  200  kgm.  acting  on  the  circumference 
of  a  wheel  of  1  m.  radius  imparts  in  30  sec.  a  speed  of  two  revolu- 
tions per  second.  Find  /  for  the  wheel. 


84  MECHANICS 

2.  Find  what  part  of  the  kinetic   energy  of  a  rolling  solid 
cylinder  is  energy  of  translation,  and  what  part  energy  of  rotation. 

The  latter  energy,  viz.,  J/w2,  can  in  this  case  be  expressed  in  terms 
of  the  mass  M  of  the  cylinder  and  its  velocity  of  translation  v.  (See  85.) 
A  simple  relation  exists  between  v  and  w. 

3.  Solve  Problem   2  for  a  rolling  hoop;    for  a  rolling  solid 
sphere. 

4.  Find  what  relation  exists  between  the  velocities  acquired  by 
a  solid  cylinder  in  sliding  without  friction  down  an  inclined  plane 
and  in  rolling  without  slipping  down  the  plane. 

Equate  in  each  case  potential  energy  at  top  to  total  kinetic  energy  at 
bottom. 

5.  The  wheel  used  in  the  falling  body  machine  (Ex.  II)  has  a 
radius  of  5  cm.  and  moment  of  inertia  of  275  gm.  cm.2  *     Find 
what  mass  at  the  circumference  would  offer  the  same  resistance 
to  an  accelerating  force  and  therefore  what  number  of  grams 
should  be  added  to  the  mass  of  the  frame  in  order  to  allow  for  the 
presence  of  the  wheel. 

6.  Find  as  in  Problem  4  the  relation  between  the  velocities  of 
solid  spheres  sliding  and  rolling  down  an  inclined  plane. 

See  equation  (86)  and  the  suggestion  under  Problem  4. 

7.  A  bullet   weighing  5  gm.   and   moving 

with  a  velocity  of  100  meters  per  second  in  the       ^ 

direction  ab    (see   Fig.    50),    strikes  the   pro- 

jection  b  of  a  fixed  wheel  whose  moment  of 

inertia  is  200,000  gm.  cm.2,*  and  whose  radius 

is  20  cm.     Find  the  number  of  revolutions  per  FIGURE  s 

second  communicated  to  the  wheel. 


*  In  order  to  understand  the  meaning  of  this  symbol  it  will  be  neces- 
sary to  consider  briefly  the  origin  and  use  of  dimensional  formulae.  The 
dimensional  formula  for  any  quantity  is  simply  the  symbol  which  repre- 
sents the  way  in  which  the  fundamental  units,  i.e.,  the  units  of  Mass, 
Length  and  Time,  enter  into  the  definition  of  that  quantity;  e.  g.,  note 
the  following  dimensional  formulae: 

L 
T 

Since  acceleration  is  velocity  divided  by  time,  a  =  -77^  = 


MOMENT    OF    INERTIA  85 

Assume  inelastic  impact.  Then  if  v  represent  the  initial  velocity  of 
the  bullet,  w  the  angular  velocity  of  bullet  and  wheel  after  impact,  and 
t  the  duration  of  the  impact,  the  velocity  lost  by  the  bullet  is  evidently 
v  —  wR,  R  being  the  radius  of  the  wheel.  Hence  the  mean  force  acting 

between  bullet  and  wheel  during  the  impact  is  /  [=  ma]  =  — - — 7 • 

The  moment  of  this  force  imparts  to  the  wheel  a  mean  angular  accelera- 
tion a  such  that  at  =  w.  From  these  equations  and  that  which  defines 
I,  w  may  be  found.  * 

8.  Find  the  number  of  revolutions  per  second  which  would 
have  been  imparted  to  the  wheel  if  the  bullet  had  moved  along 
the  dotted  line  (see  Fig.  50)  and  struck  the  wheel  at  a  point  mid- 
way between  b  and  c. 

9.  A  hoop  and  a  solid  disk  of  the  same  diameter  start  down  a 
hill  together.     Which  will  reach  the  bottom  first?     Find  the  ratio 
of  their  velocities  at  the  bottom. 

Assume  in  each  case  rolling  without  slipping  and  neglect  air  resist- 
ance. See  Problem  4. 

10.  Why  can  a  heavy  man  on  a  bicycle  always  coast  faster  than 
a  light  man  on  the  same  wheel? 

In  answer  disregard  friction. 

Since  force  is  mass  multiplied  by  acceleration,  /  =  ^=^-  =  MLT~2. 

Since  moment  of  force  is  force  multiplied  by  length,  Fh  =  -=j-  =  ML2T~2. 

Since  angular  accel.  is  linear  accel.  divided  by  length,  a  =  ^  =  ^~3- 

Since  moment  of  inertia  is  moment  of  force  divided  by  angular  acceler- 
ation 1=  ML2. 

This  last  result  might  have  been  seen  at  once  from  the  fact  that  it 
has  been  shown  that  1=  2mr2.  Enough  has  been  said  to  show  the  method 
of  procedure  for  the  derivation  of  dimensional  formulae  and  the  meaning 
of  such  formulae. 

Now  when  units  have  been  defined  but  have  been  given  no  particu- 
lar names  it  is  customary  to  write  the  dimensional  formula  in  place  of  a 
name,  replacing  in  this  formula  the  general  symbols  M,  L,  T,  by  the  par- 
ticular units  of  mass,  length,  and  time  which  have  been  used  in  the  defini- 
tion of  the  unit  under  consideration.  Thus  if  it  were  desired  to  say  that 
the  moment  of  inertia  of  a  body  was  275  units  and  to  explain  that  in  the 
definition  of  these  units  the  gram  and  the  centimeter  were  taken  as  the 
fundamental  units  of  mass  and  length  and  that  these  were  involved  in  such 
a  way  in  the  definition  that  the  dimensional  formula  for  moment  of 
inertia  was  1=  M  I?  it  would  only  be  necessary  to  write  "moment  of 
inertia  of  body  =  275  gm.  cm.2" 


V    ^ 

86  MECHANICS 

11.  Will  increasing  the  weight  of  the  tires  increase  or  decrease 
the  coasting  speed  of  a  bicycle?     What  effect  will  increasing  the 
weight  of  the  frame  have  upon  the  coasting  speed? 

12.  A  clay  ball  weighing  50  gm.,  and  moving  with  a  velocity  of 
30  meters  per  second,  struck  and  stuck  to  the  end  of    

a  rectangular  bar  1  meter  long  and  5  cm.  square 
which  was  pivoted  at  its  center  of  gravity  o  (see  Fig. 
51).     If  the  weight  of  the  bar  were  5  kilos  and  the 
motion  of  the  ball  were  at  right  angles  to  the  length       FIGTJBE  51 
of  the  bar,  what  number  of  revolutions  per  second  would  be  com- 
municated to  the  bar? 
See  equation  (87). 


XI 

SIMPLE   HARMONIC   MOTION 

Theory 

The  two  very  simple  forms  of  motion  thus  far  con- 
sidered,  viz.,  uniform  and  uniformly  accelerated,  belong 
to  the  general  class  of  non-periodic  motions. 
Among  periodic  motions  the  simplest  and  most  important  type 
is  so-called  simple  harmonic  motion.     This  is  denned  as  motion  in 
which  the  oscillating  body  is  at  every  instant  urged  toward  some 
natural  position  of  rest  with 

a  force  which  varies  directly        _  5  _  v_Jsl_       c 
as  its  distance  from  that  posi-      a  o^  —  •*•  —  5 

tion.   Thus  suppose  a  particle  FIGURE  52 

to  be  moving  back  and  forth 

over  the  path  ab  (see  Fig.  52),  under  the  action  of  a  force  which 
has  its  origin  in  o,  and  let  the  law  of  action  of  the  force  upon  the 
particle  be  expressed  by  the  equation, 


£1    m«A      *  (89) 

/2\    maj      d2 


in  which/j  and/2  represent  the  forces  acting  upon  the  particle 
when  it  is  at  the  distances  di  and  d%  respectively  from  o,  and  a^  and 
az  represent  the,  accelerations  toward  o  of  the  particle  when.it  is  at 
these  distances.  The  last  equation  may  be  written  in  the  form, 

flj    _     «2    _     «s  _ 

.       —         .       —         7       tJL^.    , 

dl       d2       ds 

or,  in  general,  if  this  constant  ratio  of  the  acceleration  to  the  dis- 
placement be  denoted  by  the  symbol  k,  the  equation  which  defines 
simple  harmonic  motion  [see  (89)]  may  be  written  in  the  form, 

a  =  Ted.  (90) 

Since/  =  ma,  (90)  may  evidently  be  written  in  the  form, 

f=mkd.  (91) 

87 


88  MECHANICS 

Any  one  of  equations  (89),  (90),  or  (91)  may  be  taken  as  the  defi- 
nition of  simple  harmonic  motion.  These  equations  express  sim- 
ply a  proportionality  between  force  and  displacement.  But,  in 
view  of  a  very  simple  relation  which  exists  between  the  character- 
istic constant  k  of  the  motion  and  the  period  of  vibration  of 
the  system,  it  is  possible  to  express  the  characteristic  equation 
of  simple  harmonic  motion  in  still  another  form.  The  evalua- 
tion of  k  in  terms  of  the  period  will  be  made  in  three  steps,  as  fol- 
lows: 

1.  The  first  step  will  consist  in  finding  Ic  in  terms  of  the  half 
length  R  of  the  path  of  the  particle,  the  velocity  v  which  it  has  at 

any  particular  point  of  this  path,  for  example  at  c,  and 
o/™#r<mdd  ^e  distance  d  of  the  chosen  point  c  from  o.  Since  the 

particle  comes  to  rest  at  the  end  of  its  path,  i.  e.,  at  #, 
the  velocity  v  is  acquired  while  it  is  moving  from  b  to  c  under  the 
action  of  the  force  which  has  its  origin  in  o.  Hence,  by  equation 
(30),  p.  43,  the  kinetic  energy  which  the  particle  has  acquired 
when  it  reaches  c,  viz.,  %mv9,  is  equal  to  the  work  done  by  the 
force  in  moving  it  from  b  to  c.  Now  it  maybe  seen  from  (91)  that 
the  value  of  the  force  acting  upon  the  particle  when  it  is  at  b  is 
inkR)  while  its  value  at  c  is  mJcd.  Since,  by  the  definition  of 
simple  harmonic  motion,  the  force  is  always  proportional  to  the 
distance  of  the  particle  from  o,  the  mean  value  of  the  force  acting 

.,,.,.,.  .  .    mkR  +  mTcd     m,      ,. 

upon  it  while  it  is  moving  from  b  to  c  is  -  —     The  dis- 

iC 

tance  be  is  equal  to  R  —  d. 

TT  mJcR  +  mkd      ,„      7, 

Hence  fynv*  =  —  -  x  (R  -  d) ; 

Z 

or  v*=k(R*-(F).  (92) 

2.  The  second  step  will  consist  in  expressing  Jc  in  terms  of  R 
and  the  speed   8  with  which  a  shadow  cast  h_ 

by  the  particle  moves  about  the 
o/Band*s      circumference  of  a  circle  drawn 

upon  ab  as  a  diameter.  Imag- 
ine that  a  distant  source  of  light,  situ- 
ated  directly  below  ab  (Fig.  53),  casts  a  FIGURE  53 

shadow   of   the  moving   particle   upon   the 
circumference  alib.     If  the  velocity  v  of  the  particle  at  any  point 


SIMPLE    HARMONIC    MOTION  89 

c  be  represented  by  the  line  ce,  then  the  velocity  S  of  the  shadow 
upon  the  circumference  is  represented  by  c'e' ' .     But 

ce        . 
-r-f  =  sin  6. 

c  e 

Hence  v  =  S  sin  0. 

Or  vz  -  S2  sin2  6  =  S2  (1  -  cos2  0)  =  Wl  -  ^Y 

Or  <- |r  •(&•!*).  (93) 

From  (92)  and  (93)  there  results  at  once 

S2 


This  equation  shows  that  the  shadow  travels  with  uniform  speed 
about  the  circumference,  for  the  expression  for  8  is  independent 
of  d.  It  involves  only  the  two  constants  k  and  R. 

3.   Since  the  shadow  has  a  constant  speed,  the  time  which  it 
requires  to  move  over  the  semicircumference  bha  is  the  length  of 

Kin  terms       this   path   divided   by  the   speed,  viz.,  —^-     But  this 

of  the  half-  S 

time  is  manifestly  the  time  t  which  the  particle  con- 
sumes in  moving  from  1)  to  «,  i.  e.,  it  is  the  half -period  of  the 
vibration.  Thus, 

<-»§••  (95) 

From  (94)  and  (95)  it  follows  at  once  that 

k  =  ^-  (96) 

Characters-  Hence  the  characteristic  equation  (90)  of  simple  har- 

tic  equations 

of  S.H.M.       monic  motion  becomes 

:  «  =  £*  '  (97) 

and  (91),  which  simply  represents  the  combination  with  (90)  of 
the  equation  f=  ma,  becomes 


^  is  called  the  force  constant  of  the  system  considered. 


90  MECHANICS 

If  it  is  known,  t  can  at  once  be  determined.     Thus,  if  the  oscilla- 
tory motion  is  due  to  the  elasticity  of  a  spring,  the 

The  force  f 

constant.        force  constant  4   can  be  determined  once   for   all   by 

d 

observing  the  force  required  to  stretch  the  spring  through  a  given 
distance.  Then  the  period  of  oscillation  of  the  system  when  any 
mass  m  is  hung  from  the  spring  can  be  calculated  from  (98). 

The  enormous  importance  of  simple  harmonic  motion  in  the 
study  of  Physics  arises  in  part  from  the  fact  that  all  vibrations 
arising  from  the  elasticity  of  matter  are  cases  of  simple 
nature'  *n  harmonic  motion.  For,  by  Exs.  VIII  and  IX,  the  restor- 
ing forces  called  into  play  by  any  sort  of  strains  in 
material  bodies  are  proportional  to  the  displacements,  so  long  as 
the  limits  of  perfect  elasticity  are  not  exceeded.  Thus  the  vibra- 
tions of  weights  suspended  from  springs,  the  vibrations  of  tuning 
forks,  of  the  strings  of  any  stringed  instrument,  of  masses  vibrat- 
ing under  the  torsion  of  a  wire,  are  all  cases  of  simple  harmonic 
motion. 

A  body  oscillating  about  an  axis  has  simple  harmonic  motion 
of  rotation    when   each   of    its   particles 
,    .      moves  with  linear  simple  har- 

Cnaracteris-  A 

ofCfQj^Mno/  monic  motion.      Thus  let  the 
rotation.         ]ine  om  (Fig.  54)  oscillate  be-    o 
tween  the  positions  oc  and  ob,  and  let  each 

point  n  of  the  line  follow  the  equation  of 

,.  •        -I        T  •  i-  •  FIGURE  54 

linear    simple     harmonic     motion,    viz., 

a  =  —d.     Since  the  linear  quantities  d  [=  be]  and  a  [=  mp]   are 
t 

related  to  the  corresponding  angular  quantities  6  and  a  by  tlic 

J  2 

equations  -^  =  0  and  -^  =  a  [R  =  om],  the  equation  a  =  -5-  d  may  be 
H  It  t 

at  once  transformed  into 

*-•£*. 


The  combination  of  this  simple  harmonic  rotational  equation  with 
the  general  rotational  relation  Fli  =  la  [see  (81)  p.  79],  gives 


SIMPLE    HARMONIC    MOTION  91 

Wh 

—£-  is  the  force  constant  of  the  rotational  system.     In  the  case 

u 

Force  con-      'n  wn^cn  the  Dotation  was  due  to  torsion  this  constant 

stant  of  Jjri 

rotation0*      T&t\o  -z-  was  given  the  name  "moment  of  torsion"  and 
u 

was  represented  by  the  symbol  T0  (see  Ex.  IX). 

It  is  evident  from  (100)  that  the  moment  of  inertia  /  of  any 
body  may  be  experimentally  determined  by  suspending  it  from  a 
iandT0from  wire  of  known  moment  of  torsion  T0  and  observing  the 
half-period  of  vibration  ^.  Thus, 

'    j  m  /  2 

or         I,±*-.  (101) 

Or  again,  if  the  moment  of  torsion  T0  of  the  suspending  wire  be 
not  known,  the  observation  first  of  ^  and  then  of  a  second  period 
t%  in  which  the  system  is  caused  to  vibrate  by  the  addition  to  1 
of  a  known  moment  of  inertia  /0,  furnishes  all  necessary  data  for 
determining  either  /or  T0.  Thus  the  half -period  of  the  system 
after  the  addition  of  I0  is  given  by 


The  elimination  of  T0  from:  (101)  and  (102)  gives 

7-^V  (103> 

Or  the  elimination  of  /from  (101)  and  (102)  gives 


Experiment 

1.   To  determine  the  force  constant  of  a  spiral  spring,  and  to 

compare  the  observed  and  calculated  values  of  the  periods  of  the 

spring  for  different  loads.     2.   To  calculate  the  periods 

of  several  "torsion  pendulums"  from  the   known  T^s 

of  the  suspending  wires,  and  to  compare  with  observations.     3.  To 

determine  /  and  T0  by  the  addition  to  a  vibrating  system  of  a 

known  moment  of  inertia  I0. 

DIRECTIONS. — 1.  First  test  Hooke's  Law  for  the  spring  and 
determine  its  force  constant  by  observing  the  elongations  produced 


92 


MECHANICS 


Force  con- 
stant of 
spiral  spring. 


by  the  successive  addition  of  100  gm.,  200  gm.,  300  gm.,  400  gm., 
beginning  with  a  load  of  50  gm.     A  graduated  mirror  placed  behind 

the  spring  (see  Fig.  55)  enables 
the  position  of  the  index  for  any 
value  of  m  to  be  accurately  de- 
termined.     In  taking  a  reading   place  the 
eye  so  that  the  image  of  the  tip  of  the  index 
is  brought   into    line  with   the   tip   of   the 
index  itself.     In  computing  the  mean  elon- 
gation per  hundred  grams  use  the  method 
outlined  on  page   76.      In    computing  the 

force  constant  -s  express  all  forces  in  dynes, 

all  lengths  in  centimeters. 

Now  replace  m   by  other  masses,  e.  g., 
150  gm.,  250  gm.,  350  gm.,and,  from  the 
force   constant   just  determined, 


m 

FIGURE  55 


Periods/or  . 

different         calculate  in  each  case  what  should 
be  the  period  of  the   suspended 


system  when  it  is  set  into   vertical  oscilla- 
tion.    Compare  these  theoretical  values  of 

the  periods  with  observed  values  obtained  by  taking  with  a  stop- 
watch the  time  of  50  vibrations. 

2.  The  torsion  pendulum  here  used   con- 
sists of  a  large  disk  suspended  as  in  Fig.  56 

from  one  of  the  wires  whose  force 

Periods  of 

torsion  pen-  constant  (moment  of  torsion)  was 
found  in  Ex.  IX.  First  find  the 
weight  of  the  disk  by  means  of  the  platform 
scales,  then  measure  its  diameter  and  compute 
its  moment  of  inertia  /  [see  (85)  p.  80].  From 
/  and  the  value  of  T0  found  in  Ex.  IX  com- 
pute the  period.  Compare  this  with  the 
observed  period  obtained  by  taking  with  a  stop- 
watch the  time  of  25  vibrations.  In  setting  the 
disk  into  oscillation,  do  not  twist  the  wire 
through  an  arc  of  more  than  5°  or  10°. 

3.  The  determination  of  T0  by  the  addition 

of  a  known  moment  of  inertia  I0  is  accomplished  FIGURE  56 


SIMPLE    HARMONIC    MOTION  93 

by  adding  to  the  plate  P  (Fig.  56)  the  large  brass  ring  D,  and 
observing  the  new  period  t2.  T0  is  then  given  by  (104).  I0  can 
TO  and  10  of  course  be  easily  calculated  from  its  mass  and  mean 
moment  ofd  diameter  [see  (84)  p.  80].  Care  must  be  tak^h  to  make 
inertia.  the  wire  pass  through  the  center  of  the  ring ;  otherwise 
the  calculated  value  of  I0  will  be  incorrect.  Determine  in  this 
manner  T0  for  each  of  the  three  wires  used  in  Ex.  IX,  and  then  cal- 
culate from  each  wire  the  coefficient  of  rigidity  n  of  steel.  [See 
(80)  p.  75]. 

From  the  same  observations  calculate  with  the  aid  of  (103)  the 
moment  of  inertia  of  the  brass  disk,  and  compare  with  the 
theoretical  value  (see  85). 


Record 


Added      Scale 
W'ts       Read's  Dif's 

0  |  _ 

100 


200 
300 
400 


mean  d  per   100  gm.  = 
force  cons't  of  spring  = 


Masses       £(obs.)        £  (calc. )       %  error 
150 
250 


2.  Mass  of  disk  P=—  Radius  =  — 
Wire  1,  To  (from  IX)  —       -   .  '.  t  x  calc.  —       -  1  1  obs.  — 
Wire  2,  T0  (from  IX)  -       -  .  '.  t  x  calc.  -       -  1  1  obs.  - 
Wire  3,  T0  (from  IX)  —       -  .  •.  1  1  calc.  —    —  t  x  obs.  - 

3.  Mass  of  ring  D  =  —   --     Mean  radius  =  — 

Wire  1,  t2  =  —  —  .  •.  To  =  ---  .  •.  n  =  --  %  dif.  from  mean  = 
Wire  2,  t.z=—  —.-.T0  =  --  .-.«  =  —  —  "  "  "  "  = 
=  -  -.-.  T0  =  -  -.:n  =  -  -"  "  "  "  = 


/  from  1  —   —    from  2  --     from  3  —  —  —    Mean  —    —    %  error  -- 

Problems 

1.  Within  a  solid  sphere  of  uniform  density  the  force  varies 
directly  as  the  distance  from  the  center.     If  the  earth  were  such  a 
sphere,  and  if  a  hole  passed  completely  through  it  along  a  diam- 
eter,  how  long  a  time  would  be  required  for  a  body  dropped 
through  the  hole  to  reach  the  other  side? 

Take  the  radius  of  the  earth  as  4000  miles. 

2.  A  horizontal  wire  one  meter  long  clamped  at  both  ends  is 
set  into  vibration  in  a  vertical  plane.     The  amplitude  at  the  mid- 


94  MECHANICS 

die  is  4  mm.  Find  the  shortest  period  which  is  permissible  if 
the  rider  at  the  mid-point  is  at  no  instant  to  lose  contact  with 
the  wire. 

3.  Show  that  the  apparent  motion  of  a  bright  point  on  the  rim 
of  a  distant  wheel,  rotating  at  uniform  speed  about  an  axis  at 
right  angles  to  the  observer's  line  of  sight,  is  simple  harmonic. 


XII 
DETERMINATION   OF    "0" 

Theory 

Let  Fig.  57  represent  any  irregular  body  which  is  oscillating 
Moment  of      un(^er  the  action  of  the  force 
force  restor-    of  gravity  about  a  horizontal 

my  a  pen-  <* 

duium.  axis  at  o.     Let  ?  be  the  dis- 

tance from  o  to  c  the  center  of  gravity 
of  the  body. 

In  order  to  find  the  law  which  gov- 
erns the  motion  of  the  body  it  is  neces- 
sary to  express  the  moment  of  force  Fh 
which  is  acting  upon  the  body  at  any 
instant  in  terms  of  the  angular  dis- 
placement at  that  instant  from  the 
position  of  rest.  If  M  be  the  mass 

of  the  body,  the  total  force  acting  upon  it  is  Mg  dynes,  and  this 
force  is  applied  at  the  center  of  gravity  c  (see  Ex.  IV,  pp.  34, 
35).  The  moment  of  this  force  is  therefore  Mg  x  dc.  But  since 
dc  =  I  sin  0,  it  follows  that 


Mg 

FIGURE  57 


Fh  =  Mgl  sin  0. 
This  equation  shows  that  the  motion  is 


The  period 


duium. 


(105) 

not  simple  harmonic, 
for  the  restoring  moment  Fh  is  not  proportional  to  the  displace- 
ment  0»  but  to  sin  0.  Nevertheless,  as  0  approaches 
zer°'  s^n  ^  approaches  0.  Hence  in  the  limit,  i.  e.,  in 
the  case  of  vibrations  of  infinitely  small  amplitude, 
pendular  motions  follow  the  law  of  simple  harmonic  motion.  The 
simple  harmonic  formula  may  then  be  applied  to  pendulum  problems 
if  only  the  arc  be  kept  so  small  that  the  error  introduced  by  the 
approximation  sin  0  =  0  is  smaller  than  the  necessary  observa- 
tional errors  of  the  experiment.  This  means  that  in  the  following 
experiment  6  should  not  exceed  5°.  Under  these  conditions, 

95 


96  MECHANICS 

then,  the  pendulum  formula  is 


J71  7 

i.  e.,  the  fcjrce  constant  of  the  motion,  viz.  —^-i  is  equal  to    Mgl. 
Substitution  in  the  S.  H.  M.  formula  (100)  gives,  then, 


This  is  the  general  formula  for  the  compound  pendulum.  If 
Period  of  the  pendulum  be  merely  a  particle  suspended  from 
pendulum.  &  weightless  thread  of  length  Z,  then 

/  (=  2wr2)  =  M  l\  (107) 

Therefore,  for  such  a  pendulum,  (106)  becomes 

i.  (108) 

^7 

The  length  of  a  compound  pendulum  is  defined  as 
^6  length  of  the  simple  pendulum  which  has  the  same 
period. 

The  center  of  oscillation  of  a  compound  pendulum  is  the  posi- 
tion of  the  particle  which  is  oscillating  naturally,  i.  e.  ,  just  as  it 
would  oscillate  if  it  alone  were  suspended  as  a  simple  pendulum 
from  the  point  of  support. 

The  radius  of  gyration  Tc  of  any  rotating  body  is  the  distance 
from  the  axis  at  which  the  whole  mass  M  might  be  concentrated 
without  changing  the  value  of  the  moment  of  inertia  /.  Thus  the 
equation, 

I=^mr*  =  Mlc*  (109) 

defines  the  radius  of  gyration  &. 

Experiment 

To  find  g  by  determining  the  length  and  the  period 
of  a  simple  pendulum. 

A  simple  pendulum  is  chosen  for  the  determination,  because 

the  length  of  such  a  pendulum  can   be   measured  directly,  while 

the  length   of    a   compound   pendulum   is   not   easily 

theorydin      obtainable.     The  time  measurement  consists  in   com- 

paring, by  the  method  of  coincidences,  the  period  of  the 

unknown  simple  pendulum  with  that  of  a  compound  pendulum 

of  known  period.     The  electric  circuit  of  the  battery  B  (see  Fig. 


DETERMINATION    OF    "</" 


97 


58)  is  completed  through  the  electro-magnet  E,  the  contact 
points  c  and  d,  and  the  two  pendulums  A  and  C.  The  length  of 
the  simple  pendulum  A  is  adjusted  until  its 
period  is  nearly  the  same  as  that  of  the 
known  pendulum  C.  When  both  pendulums 
strike  the  mercury  contacts  at  precisely  the 
same  instant,  the  click  of  the  sounder  (or 
the  stroke  of  the  bell)  E  is  heard.  Since 
the  pendulums  have  slightly  different  pe- 
riods, no  further  sound  is  heard  until  the 
faster  pendulum  has  gained  one  half-vibra- 
tion upon  the  slower,  when  the  conditions 
are  right  for  another  click.  Theoretically,  an 
observation  of  the  interval  elapsing  between 
these  two  successive  coincidences  is  sufficient 
for  the  determination  of  the  half-period  t  of 
the  unknown  pendulum.  For,  division  of 
the  time  interval  between  coincidences  by  the 
half -period  of  C  gives  the  number  of  vibra- 
tions made  by  C  during  the  interval.  The 
addition  (or  subtraction)  of  1  to  this  num- 
ber gives  the  number  of  half-periods  of  A 
during  the  same  interval.  Hence  the  divi- 
sion of  the  length  of  the  interval  by  this 
number  gives  the  half-period  sought,  viz.  /. 
However,  on  account  of  the  difficulty  of 
observing  accurately  the  exact  instant  of  a 
coincidence,  it  is  preferable  to  observe  the 
time  interval  between  two  coincidences  which 

area  considerable  distance  apart,  e.  g.,  the  interval  between  the 
1st  and  the  30th  coincidence.  The  calculation  is  then  made  pre- 
cisely as  outlined,  save  that  the  1  is  replaced  by  29. 
wac$cen  ^k*8  re(^uces  the  Observational  error  to  ?\th  its  former 
value.  It  is  not  necessary,  however,  to  watch  the 
pendulums  through  30  coincidences.  For,  if  the  interval  between 
two  successive  coincidences  be  somewhat  accurately  observed,  the 
number  of  coincidences  which  have  occurred  within  any  longer 
interval  bounded  by  two  coincidences  must  evidently  be  the 
nearest  whole  number  obtained  by  dividing  the  long  interval  by 


FIGURE  58 


98  MECHANICS 

the  time  between  two  successive  coincidences.  The  divisor,  how- 
ever, must  be  determined  with  such  exactness  as  to  remove  all 
uncertainty  as  to  what  that  whole  number  is. 

The  difficulty  of  determining  the  exact  instant  of  a  coincidence 
is  increased  by  the  fact  that,  on  account  of  the  finite  width  of  the 
contact  points  c  and  (7,  the  clicks  will  often  continue  through  a 
number  of  swings.  The  mean  time  between  the  first  and  last 
click  is  then  taken  as  the  instant  of  coincidence. 

First  very  carefully  adjust  c  and  d  until,  when  A  and  C  are  at 
rest,  each  pendulum  point  touches  the  middle  of  its  contact. 
Then  make  battery  connections  as  in  the  diagram. 
Next  by  means  of  a  thread  tie  back  pendulum  A  a 
distance  of  about  2  inches  and  set  it  into  vibration  by  burning  the 
thread.  If  a  click  is  heard  at  every  passage,  set  the  heavier  pen- 
dulum (7  into  vibration,  giving  it  an  amplitude  somewhat  smaller 
than  A's.  For  the  determination  of  t  use  an  ordinary  watch,  but 
not  one  which  has  an  error  of  more  than  10  sec.  per  day.  Take,  as 
accurately  as  possible,  the  times  of  the  first  four  coincidences; 
then  allow  the  pendulum  to  swing  for  20  or  30  minutes  and  take 
the  time  of  another  coincidence.  During  the  interval  watch  the 
pendulums  very  carefully  to  see  which  is  the  faster,  observing 
just  after  a  coincidence  and  noting  which  reaches  the  end  of  its 
swing  first.  The  length  of  A  may  be  taken,  without  appreciable 
error,  as  the  distance  from  the  knife-edge  n  to  the  center  of  the 
ball.  Leaving  the  pendulum  in  place,  measure  the  diameter  of 
the  ball  with  the  vernier-calipers  (see  Appendix),  and  the  distance 
from  the  top  of  the  ball  to  the  knife-edge  with  a  meter-stick. 


DETERMINATION  OF    "0"                                           99 

Record 

h.       m.     s.  Coinc.  perfect  at 

1st  coincidence  began  at h.      ra.       s.         Interval 

ended  at — )      m.      s. 


3d  began  at  — 

"  "  ended  at 

3d  "  began  at 

"  ended  at  - 

4th         "  began  at  - 

"  ended  at 


Last        "  began  at  — 

ended  at  - 

Interval  bet.   1st  and  last  -  No.  coincidences  in  interval  =  — 

No.  vib's  of  C  in  interval  =  -  No.  vib's  of  A  = .'.  t  =  — 

Length  of  wire  —  -  Diam.  of  ball  —  -  .  •.  length  of  pend.  —    -  .\g  =  — 

Problems 

1.  The  moment  of  inertia  of  a  long  and  thin  cylindrical  body, 
oscillating  about  one  end,  is  ^MD,   L  being 

the  length  of  the  body  and  M  its  mass.  Find 
the  radius  of  gyration  of  the  body  in  terms  of 
L.  Find  the  length  of  the  simple  pendulum 
which  has  the  same  period. 

See  (109)  and  (106).  FlGUBE  59 

2.  A   rigid    pendulum   oscillating   about  a 

horizontal  axis  has  a  period  of  .75  sec.     Find  its  period  when  the 
axis  is  inclined  45°  to  the  horizontal.     (See  Fig.  59.) 

3.  The  period  of  a  pendulum  at  the  sea  level  was  1.002  sec. 
It  was  carried  to  the  top  of  a  mountain  and  the  period  found  to 
be  1.005  sec.     Find  the  height  of  the  mountain. 


XIII 
THE   LAW   OF   CENTRIPETAL   FORCE 

Theory 

Acceleration  has  been  defined  as  rate  of  change  of  velocity.     But 

velocity  is  a  directed  quantity  (a  so-called  vector)  and  may  change 

either  in  direction,  or  in  mere  numerical  value,  or  in 

correspond-     both.     The  term  speed  has  been  agreed  upon  to  denote 

ing  to  change    , .  .'»•-»  ,T  ,.  / 

in  speed         the  numerical  value  of  the  velocity.     If,  then,  (case  1), 
the  velocity  of  a  body  at  any  time  is  represented  by  the 
line  0?*!  (see  Fig.  60),  and  if  its  velocity 
after  a  lapse  of  t  seconds  is  represented 
by  0r3,  then  there  has  been  no  change  in 
FIGURE  60  direction,  but   only   a  change   in  speed, 

and  the  mean  acceleration  has  been  -—•  The  expression  for  the 

t 

acceleration  at  the  instant  at  which  o?'!  represents  the  velocity  is 
evidently  the  limit  of  this  quantity  as  t  approaches  zero;  thus, 

(110) 

=  0. 

But  if,  (case  2),  after  the  lapse  of  the  t  seconds,  the  velocity  is 

represented  not  by  or3  but  by  or2,  a  line  equal  in  length  to  or^  then 

there  has  been  no  change  in  speed,  but  only  a  change  in 

Acceleration  .,  •       •    i          *  •••  <• 

correspond-     direction,    and,    bv    the   principle    of    composition   of 

ing  to  change 

indirection  directed  quantities  (see  lix.  Ill,  p.  22),  the  new  velocity 
orz  is  equivalent  to  the  two  simultaneous  velocities  orx 
and  TVs,  i-  e->  to  the  old  velocity  oi\  plus  a  new  velocity  ?vy  ?Va 
is  then  the  line  which  represents  the  gain  in  velocity  during  the 
time  t.  Hence  the  mean  acceleration  during  the  interval  t  has  been 

the  value  of  the  acceleration  at  the  instant  at  which  orl 


represents  the  velocity  is 


«-(-£)  CUD 

t    /t=o. 
100 


THE    LAW    OF    CENTRIPETAL    FORCE 


101 


Since,  as  t  approaches  zero,  rxr2  becomes  more  and  more  nearly  per- 
pendicular to  orl5  it  is  evident  that  in  the  limit  represented  by 
(111)  the  acceleration  a  is  at  right  angles 
to  the  velocity  oi\.     No  win  Ex.  II  force 
was  defined  as  that  which  changes  the  mo- 
tion of  a  body,  and,  by  the  Second  Law, 
whether  <  the  change  is  one  of  direction 
or  of  speed,  the  measure  of  the  force  is 
always  ma,  and  the  direction  of  the  force 
is  the  direction  of 'the  vector  a.  Thus,  in 

case  1 ,  the  force  ma    =  m  (  -y-3 \          acts 

in  the  direction  of  the  velocity,  i.  e.  in  the 
direction    oi\.       In    case    2    the   force 


ma\  = 


acts   in  the  direction 


(iV'g^oj  a  direction  which   is  rigorously   perpendicular  ft 
velocity  or^ 

Apply  these  principles  to  the  consideration  of  the  case  of  a 

body  m  moving  with  uniform  speed  S  upon  the  circumference  of 

a  circle.     (See  Fig.  61).    Let  oi\  represent  the  velocity 

circular         at  the  end  of  the  interval  t,  and  let  o'r,  or  the  equal  and 

ilini  inn*  ^B 

parallel  line  or^  represent  the  velocity  a"he  beginning 
of  the  interval  t.  Since  the  velocity  is  continually  changing  in  direc- 
tion, a  force/  must  continually  act,  and  since  the  direction  of  this 
force  is  always  at  right  angles  to  the  velocity  (see  preceding  para- 
graph), it  must  act  continually  toward  the  center.  Its  value  is  then 

f=ma  =  m(p~\  (112) 

But  since  S  represents  the  constant  speed  and  t  the  element  of 
time  considered,  it  is  evident  that 

o'o  =  St. 


But 


TT  /\     ty        *^     ^ 

Hence  r^\  =  0  o '  =  — jr- 

Substitution  of  this  value  in  (112)  gives 

mS* 
/-  ~  R  ' 


(113) 


102  MECHANICS 


•  Cf 

But  if  o>  represent  the  angular  speed,  then  o>  =  —  • 

H 


Hence  f  =  =  m»*R,  (114) 

an  equation  which  asserts  that  the  central  force  which  must 
be  applied  to  keep  a  body  in  a  circular  orbit  is  directly  propor- 
tional both  to  the  second  power  of  the  angular  velocity,  and  to  the 
first  power  of  the  radius  of  the  orbit. 

Experiment 

Object.  To  verify  the  law  of  centripetal  force. 

The   masses  mx  which  slip  along  the  rod  ab   (Fig.   62)  are 

attached  by  cords  which  pass  over  pulleys  in  the  case  p  to  the  slid- 

,         .    v  ing  collar  c.     The  central  force  which  is  necessary  to 

'  kesp  the  weights  moving  in  a  circle  is  represented  by 

^  the1  tension  jin  the  cards.     For  a  certain  critical  value  of  the  speed 

this  tension  is  equal  to  the  weight  of  the  collar  c.    In  order,  there- 

nii  p  mi 


FIGURE  62 


fore,  to  verify  the  law  stated  in  (114),  it  is  only  necessary  to 
measure  the  radius  R,  the  masses  ml  and  c,  and  to  observe  the  speed 
required  to  lift  c.  All  forces  must  of  course  be  expressed  in 
dynes,  all  masses  in  grams,  all  linear  distances  in  centimeters,  all 
angular  distances  in  radians. 

First  count  the  number  of  revolutions  of  the  axle  t*b  one  revo- 
lution of  the  wheel  Jc.     Then,  from  measurements  upon 
mlt  c,  and  72,  calculate  what  number  of  turns  N  of  the 
wheel  per  second  is  necessary  just  to  lift  c. 


THE    LAW    OF    CENTRIPETAL    FORCE  103 

To  obtain  an  experimental  value  of  the  same  quantity,  let  one 
experimenter  maintain  a  constant  rotation  of  k  at  such  speed  that 
the  collar  c  is  either  held  balanced  between  the  upper  and  lower 
stops,  or  else  continually  oscillates  back  and  forth  between  them 
(the  stops  are  so  arranged  that  c  is  free  to  move  through,  only 
about  1  mm.).  Then,  as  soon  as  this  constant  condition  is 
attained,  let  a  second  experimenter  take  with  a  stop-watch  the 
time  of  fifty  revolutions,  repeating  several  times  in  order  to  test 
the  accuracy  of  the  observations.  Then  change  both  m  and  R 
and  repeat.  Compare  in  each  case  the  observed  an*d  calculated 
values  of  the  speed. 

Record 

1st  value  of  mt=—  R=  —  c  =  —^-^          . '.  N calc.  =  — 

N  obs.  1st  trial  =—    -  2d  =  —  -   3d  =  —  -  mean  =^—  ft  error  =  — 

2d  value  of  ml  =  —  R  = c  = .  •.  N  calc.  = 

N  obs.  1st  trial  =—    -  2d  =  —    -   3d  = mean  =  —  -  %  error  =  — 

3d  value  of  m1=  —  R  =  —  c  =  —  .  •.  N  calc.  =  — 

N  obs.  1st  trial   = 2d  = 3d  = mean  = %  error  = 


Problems 

1.  Taking  the  radius  of   die  earth  as   6370  kilometers,  find 
how  many  dynes  of  force  are  required  to  hold  a  grarifof  mass  upon 
the  surface  (1)  at  the  equator;   (2)  in  latitude  45°.     Hence,  find 
what  would  be  the  values  of  g  at  the  equator  and  in  latitude  45° 
if  the  earth  did  not  rotate.     Also  find  how  many  times  the  velocity 
of  rotation  would  need  to  be  increased  in  order  that  bodies  at  the 
equator  might  have  no  weight. 

2.  The  radius  of  the  moon's  orbit  is  approximately  60  times 
the  radius  of  the  earth.     Calculate  the  force  in  dynes  which  must 
act  upon  each  gram  of  the  moon's  mass  in  order  to  hold  it  in  its 
orbit,  the  period  of  the  moon's  rotation  being  27  days  8  hours. 
Compare  this  result  with  the  force  of  the  earth's  attraction  upon  a 
gram  of  mass  at  the  distance  of  the  moon  as  computed  from  the 
law  of  gravitation.     It  was  precisely  this  computation  which  led 
Newton  to  assert  the  law  of  gravitation. 

3.  A  skater  describes  a  circle  of  10  meters  radius  with  a  speed 
of  5  m.  per  second.     What  must  be  his  angle  of  inclination  to  the 
vertical  in  order  that  he  may  be  in  equilibrium? 


MOLECULAR  PHYSICS 
AND   HEAT 


XIV 
BOYLE'S   LAW 

Theory 

The  elastic  properties  of  gases  were  very  early  made  the  sub 
ject  of  observation  and  speculation,  but  the  first  results  of  experi- 
ments made  for  the  purpose  of  discovering  the  exact 
w  re^a^on  wnicn  exists  between  the  pressure  exerted  by  a 
„  confined  gas  and  the  volume  which  it  occupies,  were 
published  by  the  English  physicist  Boyle  in  1661  in  a  work 
entitled  "Defence  of  the  Doctrine  of  the  Spring  and  Weight  of 
Air."  These  experiments  brought  to  light  the  law  which  has  since 
been  called  Boyle's  Law;  sometimes  also  called  Mariotte's  Law. 
This  law  asserts  that  so  long  as  the  temperature  remains  constant, 
the  pressure  which  a  gas  exerts  upon  the  walls  of  the  containing 
vessel  is  directly  proportional  to  its  density  or  inversely  propor- 
tional to  the  volume  which  it  occupies  ;  symbolically, 


F8 


, 


or  P!  F!  =  Pz  F2  =  P3  F3  =  etc.  =  constant.  (116) 

The  French  physicist  Mariotte  independently  discovered  the  same 
law  fifteen  years  later. 

Before  the  discovery  of  Boyle's  Law,  in  fact  before  the  begin- 
ning of  the  Christian  era,  two  theories  had  been  advanced  to 
account  for  the  elastic  properties  of  "air.  The  first  was  the  repul- 
sion theory,  according  to  which  the  pressure  exerted  by  confined 
air  was  attributed  to  repellent  forces  existing  between  the  mole- 

105 


106  MOLECULAR    PHYSICS    AND    HEAT 

cules  which  were  assumed  to  be  at  rest.     This  theory  was  held 
by  prominent  scientists  even  as  late  as  the  middle  of  the  nine- 
teenth century.     In  order  to  reconcile  the  theory  witli 

The  repulsion  .  .      .  J 

theory  of  JBoyle  s  Law,  it  is  necessary  to  assume  that  the  mole- 
cules repel  each  other  with  forces  which  are  inversely 
proportional  to  the  distances  between  them.  The  theory  has  now 
been  altogether  abandoned;  first,  because  such  a  law  of  molecular 
force  is  wholly  at  variance  with  all  modern  views  as  to  the  nature 
of  molecular  force;  second,  because  it  necessitates  the  conclusion 
that  the  pressure  which  a  gas  exerts  is  a  function  not  of  its 
density  and  temperature  alone,  but  also  of  the  shape  and  size  of 
the  containing  vessel,  a  conclusion  which  is  directly  contradicted 
by  experiment;  third,  because  the  fact  that  a  gas  does  not 
experience  a  rise  in  temperature  when  it  expands  into  a  vacuum, 
proves  that  no  repulsion  exists  between  its  molecules. 

According  to  the  kinetic  theory,  the  pressure  which  a  gas 
exerts  against  the  walls  of  a  containing  vessel  is  due  to  the  bom- 
bardment of  the  walls   by  rapidlv  moving   molecules 

The  kinetic          ...  _.  J  J 

theory  of  which  at  ordinary  pressures  are  so  far  apart  that  they 
exert  no  forces  whatever  upon  one  another,  and  which 
occupy  so  little  space  themselves  that  the  total  number  of  impacts 
per  second  against  the  walls  is  simply  the  product  of  the  number 
of  molecules  present  and  the  number  of  impacts  which  one  single 
molecule  would  make  if  it  were  alone  in  the  vessel  and  were  mov- 
ing with  the  mean  velocity  of  all  the  molecules.  Although  a 
crude  form  of  this  theory  is  as  old  as  Greek  philosophy,  it  can  not 
be  said  to  have  taken  definite  shape  before  about  1738,  when  it 
was  advanced  by  Daniel  Bernoulli i.  It  did  not  gain  general 
acceptance  until  the  middle  of  the  nineteenth  century,  when  the 
labors  of  Joule  in  England  and  of  Clausius  in  Germany  won  for 
it  well-nigh  universal  support. 

While  the  repulsion  theory  was  unable  to  account  for  Boyle's 
Law  without  the  aid  of  a  highly  improbable  assumption,  the 
kinetic  theory  furnishes  an  immediate  explanation  of  this  law. 
For,  manifestly,  if  gas  pressure  is  due  to  impacts  alone,  its  value 
at  any  instant  must  be  the  product  of  the  mean  force  of  each 
impact  and  the  number  of  impacts  taking  place  at  that  instant 
upon  a  square  centimeter  of  surface.  For  a  given  gas,  the  first 
factor  would  depend  simply  upon  the  mean  velocity  of  the  mole- 


BOYLE'S  LAW  107 

cules.  For  a  constant  mean  velocity,  the  second  factor  would  be 
proportional  to  the  number  of  molecules  present  in  a  cubic  centi- 
meter; i.e.,  to  the  density.  If,  then,  the  constancy  of  the  mean 
molecular  velocity  be  taken  as  the  condition  of  constant  temper- 
ature, it  follows  at 'once  from  the  kinetic  theory,  that  the  pressure 
should  be  directly  proportional  to  the  density.  This  is  Boyle's 
Law. 

Up  to  1848  Boyle's  Law  was  supposed  to  hold  rigorously  for 
the  so-called  permanent  gases.     In  this  year,  however,  the  French 

physicist  Regnault  performed  very  careful  experiments 
frrm  Boyle's    which  showed  that  for  air  at  ten  atmospheres  pressure, 

the  product  P  V  differs  by  about  one-fourth  of  one  per 
cent  from  its  value  at  one  atmosphere.  At  higher  pressures,  the 
departure  is  more  marked,  amounting  at  600  atmospheres  to  more 
than  25  per  cent.  These  departures  are  evidence  for,  rather  than 
against,  the  correctness  of  the  kinetic  theory;  for  when  the 
molecules  are  crowded  so  close  together  that  the  space  which  they 
themselves  occupy  is  no  longer  negligible  in  comparison  with  the 
total  volume  of  the  vessel,  then  the  kinetic  theory  would  require 
that  the. pressure  increase  more  rapidly  than  the  density;  i.e., 
that  the  product  P  V  increase.  This  is  what  actually  occurs  in 
the  case  of  all  gases  when  the  pressures  are  very  high  (100  atmos- 
pheres or  more) .  For  moderate  pressures  (1  to  50  atmospheres),  the 
departures  are  in  the  opposite  direction  in  the  case  of  all  gases 
excepting  hydrogen  (and  probably  also  helium) ;  i.e.,  the  pressure 
increases  less  rapidly  than  the  density,  or,  in  other  words,  PV 
decreases.  This  is  due  to  the  fact  that  the  attractive  forces 
between  the  molecules  are  not  wholly  negligible  in  comparison 
with  the  forces  of  impact. 

Experiment 

Object.  To  verify  Boyle's  Law  for  ordinary  pressures. 

The  body  of  dry  air  which  "is  to  be  experimented  upon  is  con- 
tained in  the  upper  part  of  the  graduated  tube  a  (see  Fig.  63), 
which  has  a  diameter  of  about  1  cm.     The  lower  end 
k  of  this  tube  is  beneath  the  mercury  in  the  cistern  AB. 

The  air-jacket  which  surrounds  a  serves  to  maintain  a  constant 
temperature  throughout  the  experiment.  If  it  is  desired  to  take 


108 


MOLECULAR    PHYSICS    AND    HEAT 


4 


T 
h 

LL 


still  further  precautions,  the  upper  part  of  AB  may 
be  filled  with  water,  although  this  is  generally  found 
to  be  unnecessary.  The  tube  a  is  graduated  in  cc. 
so  that  the  volume  V  may  be  obtained  directly  by 
reading  the  scale  u^on  a.  The  determination  of 
the  pressure  P,  which  corresponds  to  any  par- 
ticular  volume  F,  requires  the  observation,  first,  of 
the  barometer  height  H,  and  second,  of  the  differ- 
ence in  level  D  between  the  mercury  in  AB  and  in  a. 
This  observation  is  made  by  means  of  the  cathetom- 
eter  (see  below).  It  is  evident,  therefore,  that  if  the 
temperature  of  the  mercury  in  the  barometer  is  the 
same  as  that  of  the  mercury  in  AB,  then  P,  ex- 
pressed in  centimeters  of  mercury,  is  equal  to 
(II  -  D).  If  the  two  temperatures  are  different,  H 
must  be  corrected  by  multiplying  the  observed  height 
by  the  ratio  of  the  density  of 
mercury  at  the  temperature 
of  the  barometer  and  its 
density  at  the  temperature 
of  AB.  This  correction  is 
wholly  negligible  in  this  ex- 
periment unless  the  differ- 
ence in  the  two  temperatures 
amounts  to  more  than  5°C. 
DIRECTIONS.  —  To  deter- 
mine the  barometric  height 
H,  proceed  as  follows  :  By 
means  of  the  thumbscrew  s 
[see  Fig.  64,  (1)],  raise  or 

lower  the   level   of   the   mercury   in  ^hc 

cistern  E  of  the  barometer  until  the  ivory 

point  n  just  touches  the  mercury  surface. 

This  setting  can  be  made  with  great  accu- 
racy  by   observing   when   the 

.      J        J  f>  .          . 

image  oi   the   point  which  is 

. 

seen  in  the  mercury,  just  ap- 
pears to  come  into  contact  with  the  point 
itself.  This  point  n  is  the  zero  of  the  FIGURE  ei 


FIGURE  63 


Reading  of 


eter. 


(2; 


BOYLE'S  LAW  109 

scale  which  is  attached  to  the  upper  portion  of  the  metal  case 
surrounding  the  barometer  tube.  By  turning  the  milled  head  D, 
move  the  vernier  c  with  which  the  scale  is  provided  until  its  lower 
end  is  clearly  above  the  convex  surface  of  the  mercury.  Then 
carefully  lower  it  until  it  appears  to  be  just  in  'contact  with  the 
highest  point  of  this  convex  surface.  During  this  operation, 
keep  the  eye  in  such  a  position  that  the  back  lower  edge  of  the 
vernier  tube  seems  to  coincide  with  the  front  lower  edge.  To 
test  the  setting,  move  the  head  up  and  down,  and  see  to  it  that 
the  white  background  behind  the  barometer  never  becomes  visible 
above  the  top  point  of  the  meniscus.  Read  the  scale  and)  vernier 
(see  Appendix  for  theory  of  vernier).  The  capillary  correction 
for  the  barometer  is  to  be  obtained  from  the  Appendix  table, 
which  is  headed  " Capillary  Depression  of  Mercury." 

Adjust  the  cathetometer  (see  Fig.  65)  in  three  steps,  as  fol- 
lows: 

(1)  To  make  the  column  vertical, 

loosen  the    set    screw  s%   and  turn 

the  column  until  the  tel- 

Adjmtment 

of  the  escope  is  at  right  angles 

cathetsrmeter. 

to  the  line  connecting 
two  of  the  feet;  e.g.,  A  and  B. 
Then  bring  the  bubble  to  the  mid- 
dle by  means  of  the  leveling  screw 
in  the  third  foot  C.  Xext  rotate 
the  column  through  180°  about  its 
own  axis,  and  if,  after  rotation,  the 
bubble  is  displaced  from  the  middle, 
correct  half  of  the  angular  error  by 
means  of  the  leveling  screw  in  the 
foot  C  and  the  other  half  by  means  of 
the  screw  s  which  inclines  the  tele- 
scope. If  the  bubble  is  against  one 
end  of  the  level-tube,  it  is ,  impossi- 
ble to  know  when  just  half  of  the 
angular  correction  has  been  made., 
Hence  it  is  necessary  first  to  estimate- 
roughly  these  half-corrections,  then 
to  rotate  again  through  180°,  and  FIGURE  65 


110  MOLECULAR    PHYSICS    AND    HEAT 

again  to  correct,  and  thus  to  proceed  until  the  bubble  remains, 
upon  reversal,  somewhere  near  the  middle  of  the  tube.  The  half- 
corrections  can  then  be  read  off  accurately  upon  the  scale  on  the 
level-tube.  When  this  adjustment  has  been  made  to  such  a 
degree  of  accuracy  that  a  reversal  displaces  the  bubble  through 
perhaps  one  or  two  divisions,  turn  the  column  this  time  through 
90°,  i.e.,  until  the  level  is  parallel  to  the  line  connecting  A  and 
B,  and  bring  the  bubble  back  to  the  middle  by  turning  the  level- 
ing screws  A  and  B  equal  amounts  in  opposite  directions.  The 
column  should  now  be  approximately  vertical.  In  order  to  make 
it  accurately  vertical,  the  whole  operation  must  be  repeated 
from  the  beginning  with  more  care.  After  the  completion  of  the 
second  leveling,  rotation  of  the  column  into  any  position  what- 
ever should  not  cause  a  displacement  of  the  bubble  of  more  than 
half  of  a  division. 

(2)  To  make  the  line  of  sight  coincident  with  the  axis  of  the 
telescope,  first  focus  the  eye-piece  carefully  upon  the  cross-hairs  by 
slipping  the  former  forward  or  back  in  the  draw-tube ;  then,  by 
means  of  the  rack  and  pinion  with  which  the  draw-tube  is  pro- 
vided, focus  the  telescope  sharply  upon  the  scale  on  tube  a  (see 
Fig.  63),  which  should  be  set  at  a  distance  of  about  a  meter  from 
the  cathetometer.     If  moving  the  eye  slightly  from  side  to  side 
causes  the  cross-hairs  to  appear  to  move  at  all  with  reference  to 
the  scale,  repeat  both  of  these  focusings  until  this  parallax  effect 
is  wholly  removed.     Now  turn  the  telescope  in  its  socket  until 
one  of  the  cross-hairs  is  parallel  to  the  scale  divisions,  and  by 
means  of  the  screw  s3,  which  moves  the  whole  telescope  up  or 
down,  set  this  cross-hair  upon  some  chosen  division  of  the  scale ; 
then  rotate  the- telescope  in  its  socket,  i.e.,  about  its  own  axis, 
through  180°.     If  this  operation  changes  at  all  the  reading  of  the 
cross-hair  upon  the  scale,  correct  half  of  the  error  by  means  of  s3 
and  the  other  half  by  means  of  the  small  screw  which  is  found  at 
one  side  of  the  eye-piece  and  which  moves  the  cross-hair  across 
the  field  of  view.     Repeat  this  adjustment  until  rotation  of  the 
telescope  through  180°  produces  no  change  in  the  reading.     The 
line  of  sight  then  coincides  with  the  axis  of  the  telescope. 

(3)  To  make  the  axis  of  the  telescope  horizontal,  proceed  in 
either  one  of  the  following  ways  [method  (b)  is  generally  to  be 
recommended] :  (a)  Bring  the  bubble  to  the  middle  by  means  of 


BOYLE'S  LAW  111 

screw  s,  then  lift  the  level  carefully  from  the  telescope-tube,  turn 
it  end  for  end  and  replace.  If  this  operation  displaces  the  bubble, 
correct  half  of  the  error  by  means  of  s,  the  other  half  by  means 
of  the  screw  Sj,  which  adjusts  the  position  of  the  level-tube  in 
its  case.  The  telescope-tube,  and  hence  also  the  line  of  sight, 
will  be  horizontal  when  no  displacement  of  the  bubble  is  produced 
by  a  reversal  of  the  level,  (b)  Set  the  cross-hairs  of  the  telescope 
upon  some  point  on  a  scale  about  a  meter  distant.  Then  take 
the  telescope  out  of  its  socket,  turn  it  end  for  end  and  replace. 
Next  rotate  the  vertical  column  through  180°  and  look  again  at 
the  chosen  point.  If  the  cross-hairs  are  no  longer  upon  it,  correct 
half  the  displacement  by  means  of  the  screw  s  which  inclines  the 
telescope,  and  half  by  means  of  the  screw  ss  which  raises  it 
vertically.  The  telescope-tube  will  be  horizontal  when,  after 
reversal  and  rotation,  the  same  point  of  the  scale  comes  under  the 
cross-hair. 

When  the  cathetometer  is  in  complete  adjustment,  carefully 

loosen  the  set  screw  s^  slide  the  telescope  up  or  down  the  column 

until  the  cross-hair  is  near  the  top  of  the  meniscus  of 

Reading  the      ,,  .,  .    .  .  _    ,          -„.        rtrtX        , 

mercury  the  mercury  in  the  cistern  AB  (see  ±ig.  60),  clamp  s4 
and  make  the  final  setting  upon  the  meniscus  by  means 
of  the  fine  adjustment  screw  s3.  Then  take  the  reading  of  the 
vernier  upon  the  scale  of  the  cathetometer  column.  Next  raise 
the  telescope,  set  the  cross-hair  upon  the  top  of  the  mercury 
meniscus  in  the  tube  «,  and  take  a  second  reading.  The  differ- 
ence between  the  two  readings  gives  the  distance  D.  Take  a 
number  of  observations  of  this  height  in  order  to  see  to  what 
degree  of  accuracy  it  is  obtainable.  At  the  same  time  read 
through  the  telescope  the  volume  F  upon  the  scale  on  tube  a. 
This  reading  should  not  be  taken  either  at  the  top  or  at 
the  bottom  of  the  meniscus,  but  at  such  an  intermediate  point 
as  would  correspond  to  the  same  volume  if  the  meniscus  were 
flat.  This  point  can  be  obtained  only  by  careful  estimate. 
Starting  with  a  pressure  which  is  but  a  trifle  less  than  one 
atmosphere,  vary  the  volume  by  five  about  equal  steps  until  it 
is  as  large  as  can  be  conveniently  obtained  with  the  apparatus, 
and  take  the  five  corresponding  pressure  readings.  If  the 
barometric  height  varies  during  the  experiment,  take  this  fact 
into  account. 


112 


MOLECULAR    PHYSICS    AND    HEAT 


H       Read'gs  1st  trial 


in  AS     in  a       .-.  D      in  AB     in  a       .-.  D 


Record 

Read'gs  2d  trial    Mean   .-.  P  .  •.  PV  %  <*«• 
, A. .        j\  from 


FIGURE  66 


Problems 

1.  Make  estimates  of  the  probable  observational  errors  in  both 
P  and   V  above,  and   thence  deduce   the  maximum  permissible 
error  in  P  V.     Cpmpare  with  last  column  of  the  record. 

2.  A  level  is  in  adjustment  when  the  line  joining  the  points  upon 

which  it  rests  is  parallel  to  the  tangent 
drawn  to  the  highest  point  of  the  level- 
tube,  i.e.,  when  the  line  cd  (see Fig.  66) 
is  parallel  to  the  line  ab.  Show  why  in 
~  adjusting  a  level  which  is  out  of  ad- 
justment and  in  leveling  the  table  upon 

which  it  rests  (see  Fig.)  it  is  necessary  after  reversal  to  correct 

half   at   s   and  half   at   s'.     Hence  justify 

throughout  the  methods  used  in  adjusting 

the  cathetometer. 

3.  A  confined  body  of  air  V  is  placed 
under   a  pressure   of   P   mm.   of    mercury 
(see  Fig.  67.     P  =  H+ab).     By  means  of 
the   three-way   stopcock  s,   the   connection 
between  the  two  arms    (see   Fig.)  is   then 
shut  off  and  a  volume  v  of  mercury  drawn 
from  the  right  arm.     The  level  of  the  mer- 
cury in  this  arm   then  sinks  to  the  point 
a'.     Connection  between  the  arms  is   then 
reestablished  by  means  of  the  cock  and  the 
left  arm  lowered  until  the  level  in  the  right 
arm  is   again  at   a' .      The   pressure   upon 
the   air   in  the   bulb  is   now   found  to   be 

h  mm.    less    than  at   first.     Find    F,    the  FIGURE  e? 


BOYLE'S  LAW  113 

volume  of  the  bulb  down  to  the  point  «,  in  terms  of  v, 
P,  and  h. 

4.  In  Problem  3,  V  was  found  to  be  500  cc.     A  pow- 
der was  introduced  into  F  and  the  mercury  in  the  right 
arm  brought  back  to  its  original  height  a.    The  pressure 
was  then  found  to  be  900  mm.     250  cc.  of  mercury  were 
drawn  off  at  s  precisely  as  above,  and  the  pressure  was 
found  to  fall  to  500  mm.    Find  the  volume  of  the  powder. 

5.  A  volume  of  30  cc.  of  air  is  confined  in  the  closed 

arm  of  a  manometer  (see  Fig.  68) .  In  the  open  arm  the  mercury 
stands  60  cm.  higher  than  in  the  closed  arm.  What  will  be  the 
difference  in  the  levels  when  the  volume  of  the  air  is  reduced  to 
10  cc.?  (Bar.  Ht.  =  76  cm.) 


XV 


DENSITY    OF   AIR 

Theory 

The  existence  of  Boyle's  Law  makes  the  determination  of  the 

density  of  air  at  a  given  temperature  a  very  simple  matter,  at 

least  in  theory.     For,  let  a  globe  A  of  known  volume 

Density  of  .    .        J    . 

airfromtwo     V   containing  air  under  atmospheric  pressure  P«,  be 

pressures,  a 

weight  and  a    balanced   upon  a   beam  01   equal  arms  against   some 

Then  let  air  be  forced  into  the 


Suppose  that  the  weight 


body  PF  (see  Fig.  69). 
globe  until  the  pressure  within  it  is  P2. 
now  needed  to  balance  the  globe 
is  W  +  w.  Since  it  requires  the 
,  same  weight  to  balance  the  glass 
in  each  case,  it  is  evident  that 
w  is  the  weight  of  the  air  which 
has  been  forced  into  the  globe  in 
changing  the  pressure  from  P1 
to  P2.  If,  then,  V  represent  the 

volume  of  the  globe,  and  d\  and  d%  the  densities  of   air  corre- 
sponding to  the  pressures  Pl  and  P2  respectively,  it  is  evident  that 


FIGURE 


(117) 


But  by  Boyle's  Law, 


dl     Pl 


The  elimination  of  dz  from 
which  is  sought,  viz., 


(117)  and   (118)   gives  the  density 


wP, 


(119) 


FOP, -P.) 

If  the  counterpoise  W  consists  of  a  few  small  weights  and  a  closed 
glass  globe  of  the  same  external  volume  as  A,  any  errors  which 
would  arise  from  changes  in  the  barometric  height  or  the  tem- 
perature during  the  experiment  are  altogether  eliminated,  pro- 

114 


DENSITY    OF    AIR  115 

vided  only  that  the  temperature  at  which  the  air  was  introduced 
into  the  bulb  under  pressure  Pz  is  the  same  as  that  which  corre- 
sponds to  Plt  For,  with  this  arrangement,  the  buoyant  effect  of 
the  air  upon  both  sides  of  the  balance  is  the  same  no  matter  how 
rapidly  the  barometric  pressure  or  the  temperature  may  change. 
This  device  was  first  used  by  Eegnault  in  his  classical  determina- 
tion of  the  density  of  air. 


Experiment 

To  find  the  density  of  dry  air  under  existing  con- 
Object.  T  '  P 

ditions  ot  temperature  and  pressure. 

DIRECTIONS. — 1.  In  order  to  obtain  dry  air  for  the  experi- 
ment, first  see  to  it  that  the  stopcock  c  is  well  greased,  then 
connect  the  bulb  to  an  air  pump  through  a  calcium  chloride  drying- 
tube  in  the  manner  shown  in  Fig.  70.  A  pump  capable  of  produc- 


FIGURE  70 


ing  a  high  vacuum  is  unnecessary.  The  water  pump  of  Fig.  87 
may  be  conveniently  used.  First  close  the  pinchcock  s  and 
exhaust.  Then  very  slowly  open  s,  and  thus  fill  A  with  air  whi  h 

has  been  dried  by  passage  through  the  calcium  chloride 
?nmebuU)air  tube.  Repeat  this  operation  from  two  to  six  times, 

according  to  the  degree  of  exhaustion  obtainable  with 
the  pump.  Then,  in  order  that  the  bulb  may  assume  accurately 
the  outside  temperature,  wait  about  five  minutes  before  removing 
the  bulb  from  the  drying-tube  or  closing  the  cock  c.  Observe  the 
temperature  by  means  of  a  thermometer  hung  near  the  bulb,  then 
carefully  close  cock  c,  remove  and  weigh  as  follows : 

2.  First  remove  all  dust  from  the  bulb  and  the  pans  of  the 
analytical  balance  (see  Fig.  71)  by  means  of  the  camel's-hair 
brush  which  will  be  found  in  the  balance  case.  Then  very  care- 
fully suspend  the  bulb  from  one  of  the  hooks  c  and  place  upon 


116 


MOLECULAR    PHYSICS    AND    HEAT 


To  make 

Hiefirst 

weighing. 


the  other  pan  the  counterpoise  and  a  weight  of  a  few  grams  from 
the  hox  of  weights,  taking  pains  to  touch  these  weights  only  with 
the  pincers,  never  with  the  fingers.  Next  release  the 
pans  by  pushing  in  and  fastening  the  button  e  which 
controls  the  pan-arrest ;  then  turn  very  slowly  the  milled 
head  a  and  thus  lower  the  beam-arrest  just  enough  to  see  whether 
the  pointer  begins  to  move  to 
left  or  right;  i.e.,  whether  the 
chosen  weight  is  too  heavy  or  too 
light.  This  done,  raise  the  beam- 
arrest  immediately,  but  so  slowly 
as  not  to  endanger  the  knife- 
edges  by  the  slightest  jar;  re- 
place the  chosen  weight  by  the 
one  next  heavier  or  next  lighter, 
and  try  it  in  the  same  way.  Take 
the  utmost  care  never  to  place  a 
weight  on  the  pans  or  to  take  a 
weight  off  except  when  the  beam 
is  arrested.  Proceeding  thus, 
make  a  systematic  trial  of  the 


know 


gram  weights  until  you 
between  what  two  consecutive  numbers  of  grams  the  condition 
of  balance  must  lie;  then  try  in  the  same  way  the  milligram 
weights  in  order  of  magnitude  until  a  weight  is  found  such  that 
when  the  beam-arrest  is  completely  lowered  the  pointer  oscillates 
near  the  middle  of  the  scale  over  from  3  to  6  divisions.  A  larger 
swing  than  this  indicates  insufficient  care  in  lowering  the  arrest. 
The  rider  r  may  be  used  in  place  of  the  small  milligram  weights, 
if  desired,  but  no  attempt  should  ever  be  made  to  add  fractional 
portions  of  a  milligram  by  means  of  the  rider.  Before  taking  the 
resting  point,  raise  again  the  beam-arrest,  taking  pains  to  avoid  a 
jar  by  raising  at  a  time  when  the  pointer  is  at  the  middle  of  its 
swing,  close  the  face  of  the  balance  case  so  as  to  shut  out  all  air 
currents,  and  stop  all  swinging  of  the  pans  by  alternately  pressing 
and  releasing  the  button  e  which  controls  the  pan-arrest.  Then 
carefully  lower  first  the  pan-arrest,  then  the  beam-arrest,  and 
take  the  resting  point  Rlm  This  is  to  be  done  by  averaging  the 
mean  of  three  successive  turning  points  of  the  pointer  on  one  side 
with  the  mean  of  the  two  intervening  turning  points  on  the  other 


DENSITY    OF   AIR 


117 


side.  This  use  of  an  odd  instead  of  an  even  number  of  turning 
points  eliminates  completely  the  effect  of  damping.  The  following 
example  taken  in  connection  with  Fig.  72,  which  represents  the 
scale  s  of  Fig.  71,  will  make  clear  the  method  of  procedure: 

TURNING    POINTS 

Left 


15 


6.8 
7.1 

7.4 

Means  7.10 


Right 
11.-&7 


11.45 


FIGURE  72 


9.28 


Having  determined  the  resting  point  R^  slowly  raise  the  beam- 
arrest  when  the  pointer  is  in  the  middle  of  a  swing,  then  from  the 
absences  in  the  box  of  weights,  count  the  weights  which  have 
been  used  and  check  by  counting  again  as  the  weights  are 
replaced.  Call  the  sum  of  these 
weights  Wi.  Finally,  close  the  bal- 
ance-case, see  that  every  weight  is 
in  its  proper  compartment  in  the 
box  of  weights,  replace  the  pincers 
in  the  box,  and  place  the  latter  in 
the  drawer  of  the  balance-case. 

3.  Fill  the  bulb  with  air  under 
pressure  P2  as  follows:     Attach  it 

securely  to  the  pressure 

aPParatus  shown  in  Fig. 

73.  This  consists  of  a 
bicycle  pump  p  attached  to  an  air- 
tight jar  J,  which  is  furnished  with 
a  mercury  pressure-gauge  g  and  a 
calcium  chloride  drying-tube  t. 
Open  communication  between  the 
jar  and  the  bulb  through  the  dry- 
ing-tube, and  produce  a  difference 
of  level  of  50  or  60  cm.  in  the 
manometer  arms.  After  waiting 
about  five  minutes  for  the  com- 
pressed air  to  regain  the  tempera-  FIGURE  73 


118  MOLECULAR    PHYSICS    AND    HEAT 

ture  of  the  room,  read  simultaneously  the  two  arms  of  the 
manometer,  and  at  the  same  instant  close  the  stopcock  c.  Take 
the  temperature  by  means  of  a  thermometer  hung  near  the  bulb. 
If  this  temperature  differs  by  more  than  half  a  degree  from  that 
previously  taken,  the  first  weighing  must  be  discarded  and  another 
taken  under  the  same  conditions  of  temperature  as  those  which 
here  exist.  This  need  not  be  done  until  after  the  completion  of  4. 
4.  To  make  the  final  weighing,  place  the  bulb  upon  the  same 
scale-pan  as  before  and,  proceeding  exactly  as  in  2,  balance  it  by 
means  of  such  weights  Wt  that  the  pointer  again  oscil  - 
the  second  lates  near  the  middle  of  the  scale.  Then  take  the  new 

weighing. 

resting  point  Rz  in  precisely  the  manner  described 
above.  If  R%  coincided  exactly  with  R^  then  evidently  Wz  —  Wl 
would  be  the  weight  w  which  is  sought.  But,  in  general,  7?2  will 
not  coincide  with  R^  and  it  is  a  very  slovenly  proceeding  to 
attempt  to  make  it  do  so,  as  is  often  done,  by  repeatedly  shifting 
the  position  of  the  rider.  The  most  rapid  and  the  only  correct 
method  of  making  an  accurate  weighing  is  to  determine  the 
sensitiveness  *  i.e.,  tlie  number  of  scale  divisions  which  the  pointer 
is  shifted  ~b\j  the  addition  of  one  milligram,  and  then  to  calculate 
by  interpolation  the  exact  correction  which  must  be  applied  to 
Wz  in  order  to  bring  R2  precisely  into  coincidence  with  J?x;  this  is 
done  as  follows :  Immediately  after  finding  Rz,  add  to  the  lighter 
side  a  small  weight,  say  2  mg.  (1  mg.  if  the  balance  is  very  sensi- 
tive; this  may  be  done  by  means  of  the  rider  if  desired),  and 
take  the  corresponding  resting  point  7?3.  This  procedure  simply 
determines  the  value  in  milligrams  of  the  scale  divisions.  Thus, 
if  R2  =  10.63,  and  if,  upon  the  addition  of  2  mj.,  Rs  =  7.01,  then 

10.63  —  7  01 
the  sensitiveness  is  -    — - —     —=1.81.      From   the   two   resting 

points  RI  and  R*,  and  the  sensitiveness  S,  it  is  a  very  easy  matter 

*  Since,  on  account  of  the  bending  of  the  balance  arms,  the  sensitive- 
ness varies  with  the  load,  theory  requires  either  that  it  be  determined  at 
each  weighing,  or  else  that  a  table  showing  the  variation  of  the  sensitive- 
ness with  the  load  be  made  out  once  for  all  and  kept  for  use  with  the 
balance.  Since,  however,  with  good  balances,  it  generally  requires  a  very 
considerable  change  in  load  to  produce  an  appreciable  change  in  the 
sensitiveness,  it  is  usually  unnecessary  to  make  more  than  one  careful 
determination  of  the  sensitiveness  so  long  as  the  loads  involved  are  of 
about  the  same  magnitude. 


DENSITY    OF    AIR  119 

to  calculate  the  weight  which  would  need  to  be  added  to  or  sub- 
tracted from  Wz  in  order  that  the  pointer  might  be  brought 
exactly  to  the  original  resting  point  E^  Thus,  in  this  case,  the 
number  of  milligrams  which  would  be  required  to  move  the  pointer 

from  R,  [=  10.63]  back  to  El  [=  9.28]  is  W'^~^'^  =  .74.     This 

number  of  milligrams  must  be  added  to  or  subtracted  from  W& 
according  as  the  point  10.63  is  farther  from  or  nearer  to  the 
bulb  than  the  point  9.28.  Let  Ws  represent  the  corrected  value 
of  Wz.  Then  W3-  }\\  is  the  w  of  equations  (117)  and  (119). 
5.  To  find  the  volume  of  the 'bulb,  fill  it  either  with  water  or 
with  mercury,  and  weigh  upon  the  trip-scales.  Take  the  temper- 
ature of  the  liquid  used  and  obtain  the  density  from  a 
table  (see  Appendix).  From  the  weight  and  density  of 
the  liquid  calculate  V.  If  the  liquid  used  be  mercury, 
the  filling  can  most  easily  be  done  by  means  of  a  funnel  made  by 
drawing  down  one  end  of  a  piece  of  glass  tubing  so  as  to  form  a 
capillary  tube  long  enough  to  reach  into  the  interior  of  the  bulb. 
When  the  bulb  is  full  of  mercury,  it  must  of  course  be  handled 
with  extreme  care,  the  stopcock  being  always  left  open  so  as  to 
prevent  breakage  by  expansion.  Filling  with  water  may  be  done 
in  the  same  way  if  sufficient  pressure  be  applied  to  force  the  water 
through  the  capillary  tube.  It  may  also  be  done  by  alternately 
heating  and  cooling  the  air  in  the  bulb,  the  neck  being  kept 
under  water  during  the  cooling.  If  water  be  used,  the  bulb  will 
not  be  again  fit  for  use  until  it  has  been  thoroughly  dried  by 
repeatedly  exhausting  after  the  temperature  has  been  raised  above 
100°  C.  by  carefully  heating  with  a  rapidly  moving  Bunsen  flame. 
The  pressures  which  occur  in  the  numerator  and  denominator  of 
(119),  viz.  Pl  and  Pz—  P1?  must  of  course  be  expressed  in  the  same 
units.  If  they  are  expressed  in  centimeters  of  mer- 

Thebaropi-  ,  J  ,  •     „  ,  .   ,     ,, 

etercorrec-  cury,  the  two  columns  oi  mercury  which  they  represent 
must  have  the  same  temperature.*  It  is  evident,  then, 
that  since  P%  —  Pl  represents  the  difference  in  the  readings  of  the 
manometer  arms  at  the  temperature  of  the  room,  the  barometric 
reading  P^  should  also  correspond  to  the  temperature  of  the 

*  A  difference  in  temperature  not  exceeding  5°  C.  leads  to  a  wholly 
inappreciable  error.  It  may  therefore  be  overlooked  (see  table  of  mercury 
densities  in  Appendix). 


120  MOLECULAR    PHYSICS    AND    HEAT 

room;  i.e.,  for  use  in  (119)  the  observed  barometer  height  needs 
correction  only  for  capillarity.  But  in  the  table  of  "Densities  of 
Dry  Air"  given  in  the  Appendix,  the  pressures  all  represent 
heights  of  mercury  columns  at  0°  C.  Hence,  in  order  to  compare 
your  value  of  d±  with  Regnault's  value,  which  is  represented  by 
this  table,  the  observed  barometric  height  must  be  reduced  to 
0°  C.  This  reduction  is  made  by  multiplying  the  observed  height 
by  the  ratio  of  the  densities  of  mercury  at  the  room  temperature 
and  at  the  zero  temperature.  To  save  labor,  this  correction  is 
worked  out  for  all  ordinary  temperatures  in  the  table  in  the 
Appendix  entitled  "Reduction  of  the  Barometer  Height  toO°  C." 
(This  table  also  contains  a  slight  correction  for  the  expansion  of 
the  brass  barometer  scale.) 

Record 

Bar.  Ht.  obs'd  = corrected  for  capil'ty= Reduced  to  0°= cm. 

Temperature  of  air  corresponding  to  Pl  =  —  to  P2  =  — 

First  resting  point  [J5J  =  —  First  weight  [Wt]  =  —        — gm. 

Second    "        "       [«.]  =  -  Second    "      [W2]  =  -         -gm. 

Third       "        "       [R3]  =  —  Sensitiveness  [S]    =  — 

. !.  correction  to  be  applied  to  W2  = mg.  .  •.  W3  =  —        —  gm. 

.  •.  weight  of  air  w  introduced  into  bulb  (W9  —  WJ  — gm. 

Gauge  reading,  long  arm  = short  arm  =  —    -    .  •.  P2  —  P1  = cm. 

Wt.ofbulb  +  Hg .-.Wt.ofHg Tern.-         .-.  Den.  of  Hg — '— 

.  •.  V= . '-  dj  = Regnault's  value  = %  error  = 

Problems 

1.  A  glass  tube  open  at  one  end  is  60  cm.  long.     The  inside  is 
covered  with  a  soluble  pigment.     After  a  sea  sounding,  in  which 
the  tube  was  lowered  vertically,  open  end  down,  the  pigment  was 
found  to  be  dissolved  to  within  5  cm.  of  the  top.     The  density  of 
sea  water  being  1.03,  find  the  depth  of  the  sea. 

2.  A  tube  100  cm.  long  is  half  filled  with  mercury.     It  is  then 
inverted  in  a  cistern  of  mercury.     How  high  does  the  mercury 
stand  in  the  tube  above  the  cistern?     (Bar.  Ht.  76  =  cm. ) 

3.  A  barometer  tube  was  imperfectly  filled.     When  the  space 
above  the  mercury  contained  10  cc.,  the  barometer  indicated  70 
cm.  pressure.     When  the  space  above  was  reduced  to  5  cc.  by 


DENSITY    OF    AIR  121 

pushing  the  barometer  down  into  the  cistern,  it  indicated  69.5 
cm.     What  was  the  true  barometric  height? 

4.  Show  why  the  true  zero  of  a  vibration  which  is  gradually 
dying  down  because  of  damping,  is  not  obtained  by  taking  the 
mean  of  two  successive  turning  points,  one  on  the  right  and  one 
on  the  left. 

5.  Show  why  the  true  zero  is  obtained  by  averaging  the  mean 
of  two  successive  turning  points  on  one  side  with  the  intervening 
turning  point  on  the  other. 


XVI 
THE   MEASUREMENT   OF   TEMPERATURE 

Theory 

It  is  a  fact  of  common  observation  that  as  a  body  grows  hot  it 
increases  in  volume.  Quantitative  measurement  shows,  however, 
that  as  different  bodies  pass  through  the  same  change 
°^  temperature,  e.  g. ,  f rom  the  freezing  to  the  boiling 
point  of  water,  their  expansions  per  cubic  centimeter 
are  widely  different.  In  1787,  however,  a  Frenchman  by  the 
name  of  Charles  announced  that  all  gases  show  the  same  expan- 
sions as  they  pass  between  two  fixed  temperatures.  This  result 
was  confirmed  by  Gay-Lussac  some  twenty  years  later,  in  the  first 
series  of  experiments  which  were  sufficiently  exact  to  thoroughly 
justify  the  conclusion.  The  law  is  now  most  generally  known  by 
the  name  of  Gay-Lussac,  though  it  is  also  called  the  law  of 
Charles.  This  law,  like  that  of  Boyle,  has  been  found  by  later 
experiment  to  be  only  approximately  correct.  For  the  permanent 
gases,  however,  the  departures  are  only  slight,  as  will  be  seen  from 
the  table  on  p.  126.  Since  the  different  gases  obey  Boyle's  Law 
with  different  degrees  of  exactness,  these  departures  from  the  law 
of  Gay-Lussac  were  to  have  been  expected.  In  fact,  the  kinetic 
theory  requiies  the  result,  established  by  experiment,  that  the 
gases  which  show  the  largest  departures  from  Boyle's  Law,  show 
also  the  largest  deviations  from  the  mean  value  of  the  coefficient  of 
gaseous  expansion.  (See  C02  and  X20.) 

Direct  knowledge  of  the  temperature,  i.e.,  of  the  hotness,  of 

a  body  is  gained  only  through  the  sense  of  touch.     But  since,  as 

a  body  grows  hot  to  the  touch,  it  also  expands,  this 

Temperature.  J.&  .  '      . 

fact  of  expansion  has  been  made  the  quantitative 
measure  of  change  in  temperature,  even  when  the  change  is  too 
slight  to  be  perceived  by  the  touch.  Thus,  for  example,  the  vol- 
ume of  a  given  weight  of  iron  is  observed  first  at  the  freezing 
point  and  afterward  at  the  boiling  point  of  water;  the  increase  in 
volume  is  then  divided  into  100  equal  parts,  and  1  degree  rise  of 

122 


THE    MEASUREMENT    OF    TEMPERATURE  123 

temperature  is  arbitrarily  defined  as  any  temperature  change 
which  will  produce  an  expansion  of  the  iron  equal  to  one  of  these 
parts.  Thermometers  constructed  in  this  way  from  different 
substances  do  not  exactly  agree  with  one  another  for  temperatures 
other  than  0°  and  100°,  for  the  reason  that  the  expansions  of  the 
different  substances  are  not  generally  the  same  functions  of  the 
temperature.  Hence,  it  becomes  necessary  to  choose  arbitrarily 
some  particular  substance  whose  expansion  shall  be  taken  as  the 
measure  of  temperature  change.  The  gases  possess  peculiar 
advantages  for  this  purpose,  first,  because  all  gas  thermometers 
agree  with  one  another  (see  law  of  Gay-Lussac)  and,  second, 
because  the  kinetic  theory  gives  to  a  degree  measured  upon  a  gas 
thermometer  a  particular  physical  significance  (see  below).  For 
these  reasons  the  expansion  of  a  gas  has  been  chosen  as  the 
measure  of  temperature  change,  and  all  correct  determinations  of 
temperature  are  now  made  either  by  means  of  gas  thern*0meters 
or  else  by  means  of  other  instruments  which  have  been  standard- 
ized by  comparison  with  gas  thermometers. 

Gas  thermometers  take  two  forms:  (1)  the  constant-pressure 
form,  and  (2)  the  constant-volume  form.  The  first  consists  of  any 
constant-  arrangement  for  measuring  the  expansion  of  a  gas 
wnicn  is  keP^  under  a  constant  pressure.  For  example, 
suPPose  the  gas  to  ^e  confined  within  the  bulb  B  and 
dent  <>f  gases.  a  part  of  the  stem  cd  (see  Fig.  74)  by  means  of  a 
small  mercury  index  i  which  moves  without  friction  forward  or 
back  as  the  temperature  of 
the  bulb  rises  or  falls.  Let 
the  stem  be  open  at  d,  so 

that    the    confined    gas    is  FIGURE  74 

always  under  the  condition 

of  pressure  which  exists  outside.  One  degree  of  change  in  tem- 
perature is  then  defined,  in  the  centigrade  system,  as  any  tem- 
perature change  which,  starting  from  any  temperature  whatever, 
will  produce  in  the  confined  gas  an  increase  of  volume  amounting  to 

— -  of  the  increase  which  takes  place  when  the  temperature  of  the  gas 
100 

passes  between  the  freezing  and  the  boiling  points  of  water.  If  the 
barometric  pressure  never  varied,  the  stem  of  such  a  thermometer 
might  easily  be  graduated  so  that  the  instrument  would  read  tern- 


124  MOLECULAR   PHYSICS   A^D    HEAT 

perature  directly.  It  would  only  be  necessary  to  place  it  hori- 
zontally, first  in  melting  ice,  then  in  the  steam  rising  from  boiling 
water;  to  mark  the  positions  of  the  index  at  the  two  temper- 
atures ;  and  then  to  divide  the  increase  into  100  equal  parts.  In 
practice  a  gas  thermometer  is  never  treated  in  this  way,  for  the 
reason  that  such  a  graduation  would  give  correct  temperature 
readings  only  for  the  particular  pressure  P  (e.g.,  76  cm.)  at  which 
the  graduation  was  made.  In  general,  then,  in  order  to  make  a 
correct  determination  of  any  unknown  temperature  with  such  a 
thermometer,  it  is  necessary  to  know,  not  only  the  index  reading 
when  the  bulb  is  at  the  unknown  temperature  £,  but  also  the 
index  readings  which  correspond,  at  the  existing  pressure,  to  the 
freezing  and  boiling  temperatures.  From  these  three  readings  the 
temperature  t  is  obtained  without  any  actual  determination  of  any 
one  of  the  three  volumes,  i.e.,  the  volume  at  0°  [=F0],  the  vol- 
ume at  100°  [=Fioo],  or  the  volume  at  t°  [=FJ;  for  it  is  only 
the  volume  differences  (  Vt  —  F0)  and  (  Vm  —  F0)  which  need  be 
known.  Thus  it  is  at  once  evident  from  the  definition  of  temper- 
ature given  above,  that  from  such  observations  the  temperature 
on  the  centigrade  scale  is  given  by 

(130) 

oo 


100 

The  observation  at  100°  can  be  dispensed  with  if  the  actual  vol- 
ume F0  be  determined,  and  if  the  coefficient  of  expansion  of  the 
gas  between  0°  and  100°  be  known.  For,  the  coefficient  of 
expansion  a  between  any  two  temperatures  ti  and  tz  is  defined  as 
the  increase  in  volume  corresponding  to  a  rise  in  temperature 
from  ti  to  #2,  expressed  in  terms  of  the  volume  at  0°.  Thus  the 
following  equation  is  arbitrarily  taken  as  the  definition  of  a  be- 
tween ti  and  tz  : 

Vt  -Vt 


„  n 

(tZ  -   tl)     VQ 

The  coefficient  of  expansion  between  0°  and  100°  is  then  evidently 
given  by 

F100  —  FQ 


THE    MEASUREMENT   OF   TEMPERATURE 


125 


Now,  if  this  equation  be  combined  with  (120),  there  results 

;=Z^°-  (133) 

a  KO 

If,  then,  a  be  known,  it  is  evident  from  (123)  that  a  determina- 
tion of  the  temperature  t  may  be  made  by  means  of  the  constant- 
pressure  gas  thermometer,  by  measuring  the  volume  of  the  gas  at 
0°,  and  the  increase  in  volume  between  0°  and  t°.  Either  of  the 
characteristic  equations  of  a  constant-pressure  gas  thermometer, 
(120)  or  (123),  may  be  taken  as  the  definition  of  temperature  on 
the  centigrade  scale. 

Now  suppose  that  the  confined  air  in  the  bulb  B,  after  being 
raised  at  constant  pressure  P0  from  the  freezing  to  the  boiling 
c<m*tant-voi-  Poinfc  of  water,  i.e.,  from  F0  to  Vm,  had  been  brought 
mowietersond  back  a£ain»  while  at  the  boiling  temperature,  from  F100 
to  ^°  (see  Fi£*  ^)  ky  **ke  aPplication  of  additional  out- 
side  pressure,  the  final  value  of  which  is  represented  by 
Pj.  Then  Boyle's  Law  gives 

F100  _  P, 

To   =P.' 

If  this  equation  be  compared  with  (122),  which  defines  the  coeffi- 
cient of  expansion  a,  it  is  seen  that 


100  FO 


100  P 


(124) 


Now,  since,  in  the  whole  operation  which  has  been  considered, 
the  gas  has  been  brought  back  to  its  original  volume,  and  the  only 
resultant  change  which  has  taken  place  is  the  change  in  temper- 
ature from  0°  to  100°,  it  is  clear  that  Pl  is 
the  final  pressure  which  the  gas  would  have 
exerted  if  its  volume  had  always  remained  con- 
stant at  F0,  while  its  temperature  was  raised 
from  the  freezing  to  the  boiling  point  of  water, 
i.e.,  P!  is  the  pressure  which  the  gas  in  a 
constant-volume  thermometer  (see  Fig.  75)  would 
have  assumed  at  100°,  if  its  pressure  at  0°  was 
P0.  Call  it  henceforth  P100  and  write  (124), 


/1OK\ 


MOLECULAR    PHYSICS    AXD    HEAT 

Also  conceive  the  above  operation  to  be  replaced  by  that  of  raising 
the  arm  a  (Fig.  75)  and  holding  the  mercury  in  b  always  at  the 
same  point  F0  while  the  temperature  of  B  is  raised  from  0°  to 
100°,  an  operation  which  corresponds  to  that  usually  performed 
in  the  use  of  a  constant-volume  thermometer,,  'Now  the  expres- 

P     -P 

sion,      1 °         ,  applied    to   a    constant-voltfme    thermometer,    is 

-L \J\J  -L    0 

usually  called  the  pressure  coefficient  of  the  gas,  and  is  denoted  by 
the  letter  (3.  It  thus  appears  that  for  a  gas  which  follows  Boyle's 
Law  the  expansion  coefficient  a  and  the  pressure  coefficient  fi  are 
identical  quantities,  and  hence  that  the  equation 

*=AriA  (126) 


The  present 


thermometer. 


applied  to  a  constant-volume  gas  thermometer  ,  gives  precisely  the 
same  definition  of  temperature  as  does  (123)  applied  to  a  constant- 
pressure  gas  thermometer. 

In  practice,  the  constant-volume  thermometer  is  the  more  con- 
venient, as  well.  as  the  more  accurate  instrument.     It  is  therefore 
almost  exclusively  used.     But  there  are  other  reasons 

.  .  . 

aside  irom  those  01  convenience  and  accuracy,  wn.cn 

. 

make  a  choice  necessary.  As  has  just  been  proved,  the 
pressure  and  expansion  coefficients  must  be  identical  for  gases 
which  rigorously  follow  Boyle's  Law.  But,  since  such  gases  do 
not  exist,  it  was  to  have  been  expected  that  refined  measurements 
would  show  slight  differences  between  these  coefficients.  The 
following  table  gives  the  results  of  some  of  the  best  determina- 
tions, those  of  Chappuis  at  the  International  Bureau  of  Weights 
and  Measures  being  probably  the  most  accurate  yet  made: 


Gas                        C 
Hydrogen  . 

Expansion 
Joefflcient  a 

0036600  * 

Pressure 
Coefficient  ft 

0036625 

Observer 
P   Chappuis 

Date 

1887 

Hydrogen  

.003661 

003667 

^  Regnault 

1840 

Air  

.003670 

.003665 

Regnault 

1840 

Air 

003663 

Kuenen  and  Randall 

1895 

Carbon  dioxide  (CO2). 
Nitrous  oxide  (N2O).  . 

.003710 
.003719 

.003688 
003676 
003674 

Regnault 
Regnault 
von  Jolly 

1840 
1840 
1874 

003667 

1874 

Argon 

003668 

Kuenen  and  Randall 

1895 

Helium 

003665 

Kuenen  and  Randall 

1895 

*See  Traverse*  "Experimental  Study  of  Gases,"  p,  149.    Macmillan  &  Co.,  1901. 


THE  MEASUREMENT  OF  TEMPERATURE  127 

Because,  then,  of  the  slight  difference  in- the  two  coefficients 
for  the  same  gas,  and  because  of  the  slight  differences  existing 
between  the  different  gases  as  regards  the  same  coefficient,  the 
International  Committee  of  Weights  and  Measures  adopted  in  1887 
the  constant -volume  hydrogen  thermometer  as  the  standard  of  tem- 
perature measurement.  Hence,  the  above  approximately  correct 
definition  of  temperature  must  be  replaced  by  that  embodied  in  (126) 
when  applied  to  a  hydrogen  thermometer.  In  other  words,  since 

fi  =  .0036625  =  --,    I°C.    is    by    definition    such    a    temperature 


change  as  will  cause  the  pressure  of  a  body  of  hydrogen  which  is 
kept  at  constant  volume  to  change  by  ^—  of  its  zero  value. 

According  to  the  kinetic  theory,  the  pressure  exerted  by  a  gas 

which  follows  Boyle's  Law  is,  at  any  instant,  the  product  of  the 

mean  force  of  each  impact  and  the  number  of  impacts 

The  Kinetic 

theory  and      occurring  at  that  instant  upon  each  sq.  cm.  01  suriace 

temperature.  °  •  * 

(see  Jix.  Alv).  JNow,  the  number  01  impacts  wnicn  a 
single  molecule  which  is  moving  within  a  given  enclosure  will 
make  per  second  upon  the  walls  is  evidently  proportional  to  its 
velocity.  Hence,  with  a  fixed  number  of  molecules  present  in 
the  enclosure,  the  number  of  impacts  occurring  at  any  instant 
must  be  directly  proportional  to  the  average  velocity.  But  the 
mean  force  of  each  impact  is  also  proportional  to  the  velocity  (see 
second  paragraph  below).  Hence  with  a  given  kind  of  mole- 
cule, i.e.,  a  given  gas,  the  pressure  exerted  against  the  walls  is 
proportional  to  the  square  of  the  molecular  velocity.  But  also, 
for  a  given  kind  of  molecule,  the  kinetic  energy  (%mv2)  of  each 
molecule  is  proportional  to  the  square  of  the  velocity.  Hence  the 
ratio  of  the  two  pressures  which  a  constant  volume  of  gas  exerts 
at  different  temperatures  is  simply  the  ratio  of  the  mean  molec- 
ular kinetic  energies.  Since,  then, 

ET 

(127) 

it  follows  that  the  equation  (126)  which,  when  applied  to  a  con- 
stant volume  of  hydrogen,  defines  temperature,  may  also  be 
written 

(KE},-(KE\ 
ft  (KE), 


128  MOLECULAR    PHYSICS    AND    HEAT 

Since  ft  =  -^-,  and  since,  when  (KE)t-  (KE}Q  =  ft  (KE)^  t  =  1, 

A  I  O 

this  maybe  expressed  in  words  thus:  1°  change  in  temperature 
means  merely  the  addition  (or  subtraction)  of  a  given  amount  to  the 
mean  kinetic  energy  of  translation  of  the  flying  hydrogen  molecules  , 

and  this  amount  is  —^  of  the  kinetic  energy  which  these  molecules 


possess  at  0°  C. 

It  is  evident,  then,  that  if,  starting  from  0°C.,  this  amount 

were  subtracted  273  times,  all  kinetic  energy  would  be  removed 

from  the   hydrogen   molecules;  i.e..  they  would  come 

The  absolute  '       .    , 

zero  of  tem-  completely  to  rest.  Further,  the  interpretation  of  the 
law  of  Gay-Lussac,  in  the  light  of  the  kinetic  theory,  is 
that  the  molecules  of  all  other  gases  would  come  to  rest  at  the 
same  temperature,  viz.,  at  —  273°C.  This  point  at  which  all  molec- 
ular motion  would  cease  is  called  the  absolute  zero  of  temperature. 
Of  course  this  temperature  can  never  be  attained  ;  but  within  the 
last  twenty  -five  years  great  strides  have  been  made  toward  it.  Up 
to  1877,  the  lowest  attained  temperature  was  —  110°0.,  produced 
by  Faraday  in  1845  by  the  rapid  evaporation  in  vacuo  of  a  mixture 
of  ether  and  solid  C02.  The  temperature  of  the  solid  C02  alone 
is  —  80°C.,  that  of  the  mixture  ordinarily  about  the  same.  In  1877 
a  Swiss,  Pictet,  and  a  Frenchman,  Cailletet,  independently  lique- 
fied oxygen,  which  has  a  boiling  point  of  —  182.5  °C.  But  as 
neither  of  these  experimenters  obtained  the  liquid  oxygen  in  a 
static  condition,  they  could  make  no  observation  of  its  temper- 
ature. The  lowest  measured  temperature  was  —  140°C.,  obtained 
by  Pictet  by  the  rapid  evaporation  of  liquid  N20.  Following  these, 
the  two  Poles,  Wroblewski  and  Olszewski,  and  the  Englishman 
Dewar  accomplished  the  liquefaction  of  air  in  quantity,  and  by  its 
evaporation  in  vacuum  produced  and  measured  temperature  as 
low  as  —  210°C.  In  1885,  Professor  Olszewski  liquefied  hydrogen 
and  located  its  boiling  point  at  —243.5°.  It  has  since  been  not 
only  liquefied  in  quantity,  but  also  solidified  by  Dewar  (1900), 
according  to  whose  most  recent  determinations  the  boiling  point 
of  hydrogen,  measured  by  means  of  a  hydrogen  thermometer, 
is  between  —252°  C.  and—  253°C.  Its  melting  point  is  between 
—  256°C.  and  —  257°C.  The  lowest  temperature  obtained  by  evap- 
orating solid  hydrogen  is  —  260°C.,  only  13°  above  absolute  zero. 


THE   MEASUREMENT    OF   TEMPERATURE  129 

Thus  far  it  has  been  shown  that  1°  rise  in  temperature  sig- 
nifies an  increase  in  the  mean  molecular  energy  amounting  to 

Pressure        973  °^   ^he  zero  value  of  this  energy,  but  no  attempt 

andmplec-        *' * 

uiar  unetic     has  vet  been  made  to  find  the  actual  value  of  this  zero 

energy.  •> 

energy,  or  to  discover  whether  any  simple  relation 
exists  between  the  mean  kinetic  energies  of  the  molecules  of 
different  gases  which  have  the  same  temperature.  In  1851,  the 
English  physicist  Joule  worked  out  in  the  following  way  the  precise 
relation  which  must  exist  between  the  pressure  exerted  by  a  gas 
and  the  mean  kinetic  energy  of  the  individual  molecules. 

Let  the  gas  be  contained  in  the  vessel  mn  (see  Fig.  76),  the 
lengths  of  whose  sides  are  «,  #,  c,  and  let  N  represent  the  number 
of  molecules  in  the  vessel  and  v  the 
average  velocity  of  each  molecule.  The 
particles  are,  of  course,  moving  in  all 
possible  directions,  but,  on  the  whole, 
the  pressure  must   be    just  the  same 
as    though    one-third   of    them    were  FIGURE  76 

moving  parallel  to  each  of  the  three 

edges  a,  Z>,  c.  If  a  single  particle  were  alone  in  the  vessel  and 
were  moving  back  and  forth  parallel  to  one  edge,  e.  g.,  to  c, 

v 

it  would  make  against  the  upper  wall  —  impacts  per  second ;  for 

v  is  the  distance  which  it  moves  per  second  and  2c  is  the  distance 
moved  between  successive  impacts  against  this  wall.  If  there  are 
other  molecules  present  in  the  vessel,  the  molecule  considered 
may,  of  course,  collide  with  them  in  its  excursions  to  and  fro,  but, 
so  long  as  the  volume  occupied  by  the  molecules  themselves  is 
negligible  in  comparison  with  the  volume  of  the  vessel,  the  num- 
ber of  impacts  against  the  wall  ab  will  be  unaltered  by  these  col- 
lisions. For  (see  Ex.  VII,  Problem  1)  when  two  equal  elastic 
particles  collide,  the  effect  is  the  same  as  though  one  particle  had 
passed  through  the  other  without  influencing  it.*  Since,  then, 

1) 

each  particle  which  is  moving  parallel  to  c  makes  —  impacts  per , 

*  This  was  proved  only  for  the  case  of  central  impact  and  can  not 
therefore  be  taken  as  a  complete  justification  for  the  preceding  statement. 
However,  a  more  rigorous  analysis  than  the  one  here  given  leads  to  pre- 
cisely the  same  final  result. 


130  MOLECULAR    PHYSICS    A^D    HEAT 

JV7 

second  against  the  side  «Z>,  and  since  there  are  ^  particles  so  mov- 

o 

ing,  the  total  number  of  impacts  per  second  against  ab  must  be 

N      v 

—  x  —  •     Now,  at  each  impact  each  molecule  first  loses  all  its 
o       lie 

velocity  and  then  gains  an  equal  opposite  velocity.  Hence  the 
change  in  velocity  which  each  molecule  experiences  at  each  impact 
is  %v.  The  change  in  momentum  which  each  particle  experiences 
at  impact  is  therefore  %mv,  m  being  the  mass  of  each  molecule. 
The  total  change  in  momentum  experienced  by  the  total  mass  which 

N       v 
impinges  per  second  against  ab  is  then  2m  v  x  —  x  —  •   But  rate  of 

O  /iC 

change  of  momentum  is  the  measure  of  force  (see  Ex.  II).  Hence 
the  force  /",  which  the  w&llab  experiences  because  of  the  molec- 
ular impacts,  is  given  by 

-    n          N       v       ,  Nmv*  /int\\ 

/=2)B(,x_x_  =  i_.  (139) 

But  the  pressure  p  is  by  definition  the  force  per  unit  area.     Hence, 


But  abc  is  merely  the  volume  V  of  the  vessel.  Hence  equation 
(130)  becomes 

^r=-J-  Nmv9.  (131) 

Now,  Boyle's  Law  asserts  that  the  product  of  the  pressure  and 
volume  of  any  gas  is  constant  so  long  as  the  temperature  remains 
constant.  Hence,  according  to  equation  (131),  the  condition  for 
constant  temperature  is  a  constant  mean  velocity  of  molecular 
motion.  If  n  denote  the  number  of  molecules  in  unit  volume, 

N 
then  n  =  -^,  and  (131)  becomes 

p  =  $  nmv*.  (132) 

The  kinetic  energy  of  the  n  molecules  is  %nmv*.     Hence  the  pres- 
sure exerted  by  a  gas  is  numerically  equal  to  two-thirds  of  the 
'kinetic  energy  of  translation  of  the  molecules  in  unit  volume. 
E  uanit  of  ket  ^W0  gases  1  an(^  ^  be  contained  in  different  ves- 

se^s'    ^ut  let  them   have   the   same   temperature   and 
exert  tne  same  Pressure-     According  to  equation  (132) 
temperature.    fae  pressure  in  Vessel  1  is 

'  p  =  -3  tiiiniv*. 


THE    MEASUREMENT    OF   TEMPERATURE 


131 


Similarly  in  vessel  2, 


Now,  according  to  the  law  of  Avogadro,  the  proof  of  which  will 
be  advanced  in  the  next  section,  if  the  two  gases  exert  the  same 

pressure  and  are  at  the  same 
temperature,  they  contain  the 
same  number  of  molecules  per 
unit  volume,  i.e.,  in  this  case 
HI  =  %.  Hence, 


i.e.,  at  a  given  temperature  the 
molecules  of  all  gases  have  the 
same  average  kinetic  energy  of 
translation.  Thus  the  kinetic 
theory  leads  not  only  to  the 
conclusion  that  1°  rise  of  tem- 
perature means  an  increase  of 
kinetic  energy  amounting  to 

1 

energy, 


— —  of  the   zero 


FIGURE  77 


but 

also  that  equality  of  tempera- 
ture in  gases  means  equality  in 
the  average  kinetic  energies  of 
the  molecules. 

Experiment 

1.   To  determine  the  pres- 
sure coefficient  ft  of    air.     2. 

To     standardize    a 
Object 

"mercury  in  glass " 

thermometer  by  comparing  it 
with  an  air  thermometer. 

Fig.  77  shows 
the  constant  -  vol- 
ume air  thermom- 
eter which  is  to  be 
used.  The  volume 
is  kept  constant  by 


132  MOLECULAR    PHYSICS   AND   HEAT 

so  adjusting  the  height  of  the  right  manometer-arm  that  the  mer- 
cury in  the  left  arm  is  brought,  before  each  reading,  exactly  into 

coincidence  with  the  platinum  point  c  (see  also  Fig.  77a). 

The  fine  adjustment  screw  e  facilitates  this  setting.  The 
difference  in  the  mercury  levels  in  the  two  arms  is  read  off  upon  the 
central  graduated  scale.  A  mirror  is  attached  to  the  back  of  the 
vernier  index  i  and  slides  with  it  so  that  the  readings  can  be  made 
entirely  free  from  the  error  of  parallax.  A  three-way  stopcock  s 
makes  it  possible  to  put  the  bulb  into  communication  either  with  the 
manometer  or  with  the  outer  air.  The  form  of  this  cock  is  shown 
in  Fig.  77b,  which  represents  the  position  when  it  is  in  communica- 
tion, through  the  hole  o  and  the  tube  m,  with  the  manometer. 
Turning  the  cock  through  180°  brings  the  bulb  into  communica- 
tion, through  the  hole  r  and  the  tube  n,  with  the  outside  air. 
,  A  rigid  arm  ?,  which  is  attached  to  the  sliding  collar  (7,  holds  the 
bulb  in  place  and  protects  it  from  breakage. 

If  PQ  represent  the  observed  pressure  at  0°,  H  the  barometric 
height  and  hQ  the  difference  between  the  readings  in  the  two  arms 

when  the  bulb  is   surrounded   with   melting   ice   and 

the  mercury  is   brought   into   exact   contact   with  c, 
then  evidently 

p0=H+h0, 

JiQ  being,  of  course,  negative  if  the  level  in  the  left  arm  is  highei 
than  that  in  the  right.  Similarly  1he  observed  pressure  at  100° 
is  given  by 

pm  =  H+hm, 
and  that  at  any  unknown  temperature  t,  by 

pt  =  H+  lit. 

Now  if  the  conditions  assumed  in  the  deduction  of  (125)  and  (126) 
had  all  been  fully  realized  there  would  result  very  simply 

n         PIOO  ~  PO        PIW~PO  "100  ~  UQ        . 


100  Po        lOOjOo  ~  100  (H+  hQ) 

4        -  -          h  -  h 


,  . 


But,  in  point  of  fact,  pmi  p^  andjo,  are  all  slightly  different  from 
Aooj  PO?  and  P,,  for  the  latter  correspond  to  an  absolutely  constant 
volume  and  to  a  condition  in  which  all  of  the  confined  air  under- 
goes tho  heating  or  cooling  operation.  Since  neither  of  these 


THE  MEASUREMENT  OF  TEMPERATURE          133 

conditions  is  realized  in  practice  two  corrections  must  be  applied 
to  the  observed  pressures.     The  first  is  the  correction  for  the 
expansion   of    the   bulb.      If  y  represent  the  cubical 
coefficient  of  expansion  of  glass,  (i.e.,  the  expansion 
Per  cc-  Per  Degree,  a  quantity  which  is  equal  to  .000025), 
then  the  initial  volume  V  of  the  bulb  will  have  changed 
at  100°  to  F+  100yF[=F(l  +  100y)].      In  order  to  reduce  this 
volume  back  to  F,  it  would  be  necessary  to  apply  some  pressure 
x  such  that  (see  Boyle's  Law), 

F(1+10°Y)  or       a-p^l-HOOy).        (135) 


F- 

^100 

Applying  this  correction  then  to  both  (133)  and  (134)  there  results  : 
pm  (1  +  IQOy)  -po     pm-p<>        ypm  . 

*  woPo          "-  -" 


and          t  „.  -    =         o_.*  (137) 

£^o  Pp*-ypt 

But  for  even  moderately  accurate  results  a  still  further  correc- 

tion must  be  applied  to  all  of  the  observed  pressures  in  order  to 

make  allowance  for  that  portion  of  the  confined  air 

Correction  .  . 

for  unheated  which  escapes  the  changes  in  temperature  which  take 
r>  place  within  the  bulb.  Thus  it  is  evident  that  if  the 
air  in  the  capillary  stem  and  in  the  space  about  c  fell  to  the  zero 
temperature,  pQ  would  be  somewhat  smaller  than  it  is.  Similarly  if 
this  air  were  heated  with  the  rest  to  100°,  p^  would  have  a  larger 
value.  The  error  arising  from  this  source  is  eliminated  by  multi- 
plying the  /?  and  t  given  by  (136)  and  (137)  by  a  factor  of  the  form 

i    |    v    P  * 

V  po  1  +  .  00367*' 

in  which  v  is  the  volume  of  the  unheated  portion  of  F,  t'  the  tem- 

perature of  the  room  near  e,  and  p  is  pm  if  the  correction  is  applied 

to  /3,  pt  if  it  is  applied  to  t.    The  deduction  of  the  form  of  this  factor 

is  comparatively  easy  but  will  scarcely  be  found  profitable  here  (see 

Kohlrausch's  Leitfaden  der  Praktischen  Physik,  8th  ed.,  p.  112). 

Of  course,  if  the  barometer  height  is  not  760  mm.,  the  temper- 

ature of  steam  will  not  be  exactly  100°,  and  a  final  modification 

must  be  made  of  (136)  to  take  into  account  the  change 

Correction        .      .     ...  .  \  ,     \  . 

for  boiling      in  boiling  point  with  change  in  pressure.     The  correc- 

temperature.  ..  '  ,    ,  ,   ...  , 

tion  will  be  made  by  replacing  the  100  of  equation 
*  The  last  expression  is  obtained  by  solving  the  preceding  equation  for  t. 


134  MOLECULAR    PHYSICS    AND    HEAT 

(136)  by  the  value  of  the  boiling  point  taken  from  table  6  in  the 
Appendix. 

DIRECTIONS. — 1.  Lower  C'  (Fig.  77)  until  the  mercury  in 
the  left  arm  of  the  manometer  is  below  the  three-way  stopcock  s, 
The  pressure  then  °Pen  communication  through  r  and  n  (see  77b), 
coefficient  p.  between  B  and  the  outside  air.  Attach  to  n  a  calcium 
chloride  drying  tube  and  an  exhaust  pump  precisely  as  in  the  last 
experiment.  When  B  has  been  thoroughly  dried,*  turn  the  cock  so 
as  to  restore  communication  with  the  manometer.  Then,  with  the 
aid  of  an  ice  scraper  (see  Fig.  78),  prepare 
several  quarts  of  pure  shaved  ice,  pack 
it  carefully  about  the  bulb  within  any  con- 
venient vessel  from  which  the  water  may  be 
FIGURE  78  drained,  at  the  same  time  lowering  C'  so  that 

the  contraction  of  the  air  in  B  may  not  cause 

mercury  to  pass  over  into  the  bulb,  then  pour  over  distilled 
water  and  repack.  The  water  is  added  both  to  insure  good 
contact  and  to  make  certain  that  the  temperature  of  the  ice 
is  not  below  0°.  A  mixture  prepared  in  this  way  from  shaved 
ice  or  clean  snow  gives  a  perfectly  constant  zero,  but  dry  ice 
or  snow  can  not  be  depended  upon.  Let  the  ice  surround  the 
whole  bulb  and  the  tube  t  up  to  the  point  to  which  the  steam  will 
have  access  when  the  bulb  is  immersed  in  the  steambath.  After 
a  lapse  of  five  minutes,  adjust  C'  and  e  until  the  mercury  just 
touches  the  platinum  point  c\  then  set  the  sliding  index  i  succes- 
sively upon  the  tops  of  the  mercury  menisci  in  the  two  arms,  and 
take  the  corresponding  vernier  readings  upon  the  vertical  scale. 
Repeat  both  the  setting  and  the  observations,  and  see  how  closely 
successive  values  of  ho  can  be  made  to  agree.  Next  insert  an 
asbestos  screen  between  the  bulb  and  manometer,  replace  the  ice  by 
the  steam  jacket  shown  in  Fig.  77,  and,  as  soon  as  the  expansion  has 
ceased,  make  again  a  series  of  settings  and  readings.  In  order  to 
preclude  the  possibility  of  a  leak,  it  is  well,  during  the  expansion,  to 
keep  the  mercury  always  above  the  stopcock.  After  reading  the 
barometer,  obtain  the  volume  of  unheated  air  as  follows:  Place  a 
small  beaker  underneath  n  and  raise  C"  carefully  until  the  mercury 
rises  in  the  capillary  tube  t  to  the  point  to  which  this  tube  was 

*  Of  course  this  drying  need  not  be  repeated  with  every  experiment. 
The  instructor  will  decide  in  each  case  whether  it  is  necessary  or  not. 


THE    MEASUREMENT    OF    TEMPERATURE  135 

heated.  Then  turn  the  cock  s  and  let  mercury  run  out  through  n 
until  the  mercury  level  in  the  left  arm  has  exactly  reached  the  plati- 
num point  c.  From  the  weight  of  the  mercury  which  has  emerged 
from  5,  v  can  at  once  be  obtained.  The  volume  of  J9,  viz.,  F, 
can  be  calculated  with  sufficient  accuracy  from  a  careful  estimate 
of  the  dimensions  of  the  bulb.  Or  the  bulb  may  be  immersed  in  a 
graduate  and  the  rise  of  the  water  noted.  If  v  is  of  the  order 
y 
——  an  error  of  as  much  as  10%  in  the  measurement  of  Fwill 

introduce  into  the  determination  of  (3  an  error  of  but  one- tenth  of 
one  per  cent. 

2.  Having  found  /?,  surround  B  with  a  water  bath,  immerse 
also  in  the  bath  the  mercury  thermometer  which  is  to  be  standard- 
ized, seeing  to  it  that  the  whole  of  the  mercury  in  both 
of  mercury     bulb  and  stem  is  as  nearly  immersed  as  possible,  and 

thermometer.     ,  _    ..  . ,  .     , 

find  the  corrections  of  the  mercury  thermometer  at  about 
25°C.,  50°C.  and  75°C.  [see  (137)].     In  all  of  these   determina- 
tions,   stir  the  water   continually  and    take  no  readings 
until  a  stationary  condition  has  been  reached.     Next  find       (o) 
the  correction  of  the  mercury  thermometer  at  0°,  either 
by  filling  a  large  funnel  with  pure  shaved  ice,  pouring 
over  distilled  water  and  immersing  the  thermometer  up  to 
its  0°  mark ;  or  better  still  by  immersing  it  in  a  small  vessel 
of  distilled  water   (see  Fig.  79)   which  is  surrounded  by 
a  freezing  mixture  of  ice  and  salt,  and  noting  the  point 
at  which,  with  continual  stirring,  the  mercury  becomes 
stationary.     In  the  latter  case  it  will  generally  be  found 
to  fall  below  the   freezing  point  and  then  to   rise  sud- 
denly to  it  as  the  ice  crystals  begin  to  appear.     Lastly 
find  the  correction  at  the  boiling  point  by  immersing  the 
thermometer   up   to   the    mark   100    in  a  double-walled 
steam  jacket  like  that  shown  in  Fig.  77.     The  correc- 
tion will  be  the  difference  between  the  observed  boiling 
temperature  and  that  given  in  the  Appendix  for  the  exist- 
ing pressure.     It  will  be  positive  if   the   mercury  ther- 
mometer reads  too  low,    negative  if    it  reads  too   high.     ^^ 
In   all   readings   of  thermometers,    take   great   pains  to 
avoid  the  error  of  parallax.     This   is  done  by  placing  the  stem 
carefully  at  right  angles  to  the  line  of  sight. 


136  MOLECULAR    PHYSICS    AND    HEAT 

From  the  five  observed  points  plot  a  curve  of  thermometer 
corrections  (cf.  also  the  two-point  curve  of  Figure  86,  Ex.  XVIII). 

Record 

1.  Determination  of  /3. 

Melting  ice                1st  setting  2d        Steam  1st  setting  3d 

Reading  at  c  = 

Reading  in  rt.  arm  = 

H  =; HQ  = Tiioo  = 

Mean  7io  = . '.  Po  = Mean  hm  = . '.  PICO  =  — 

v  =  —  V=  —  .  •.  /3  =  —    —  %  error  =  — 

2.  Correction  of  mercury  thermometer.  H=  — 

0°        1st    2d         25°        1st    2d          50°        1st    2d        75°        1st      2d 
Ate 
Rt.  arm 


Means  /io htl .  •.  t± ht2 .  •.  ti  —         ht3 .  •.  1 3  — 

By  mercury  thermometer  1 1 t2 1 3  — 

Correction  of  Hg.  ther.  at  0°  —  at  25° at  50° at  75° at  100°  - 

Problems 

1.  From  equation  (126),  which  defines  tf,  the  measure  of  tem- 
perature upon  the  centigrade  scale,  show  that,  if  the  volume  of  a 
gas  remain  constant  while  the  temperature  changes, 

ph      273  +  *!. 

Ph  ~  273  +  4' 

or,  if  T  denote  the  temperature  measured  from  a  point  273°  below 
0°C.,  i.e.,  from  the  absolute  zero,  show  that 

A       T, 

pf  =  T9'  (138) 

2.  Show  also  that  if  the  pressure  of  a  gas  remain  constant 
while  its  temperature  changes,  the  volume  is  directly,  the  density 
Inversely,  proportional  to  the  absolute  temperature. 

'3.  Since  Boyle's  Law  gives  the  variation  in  density  when  the 
pressure  alone  changes,  the  temperature  remaining  constant,  it  is 
evident  that  from  this  law,  and  the  rule  deduced  in  the  preceding 
problem,  the  density  of  a  gas  can  be  calculated  for  all  temperatures 


THE  MEASUREMENT  OF  TEMPERATURE  137 

and  pressures  if  it  has  once  been  determined  for  one  single  value 
of  the  temperature  and  pressure.  Thus,  given  that  the  density  of 
air  atl6°C.,  745  mm.,  is  .001192;  find  the  density  at  0°,  745 
mm. ;  at  0°,  760  mm. ;  at  120°,  755  mm. 

4.  Find  the  volume  occupied  by  28.88  gm.  of  air  at  0°,  760 
mm. 

5.  How  much  work  is  done  against  atmospheric  pressure  when 
10  gm.  of  air  at  0°,  750  mm.,  are  heated  to  50°,  750  mm.? 

6.  In  equation  (132),  nm  is  merely  the  mass  per  unit  volume, 
i.e.,  the  density  of  the  gas.     The  density   of  air  at   0°C.,   76 
cm.,  is  .001293;    hence,  find  from  (132)  the  average  velocity  of 
the  molecules  of  air  at  this  temperature  (p  must,  of  course,  be 
expressed  in  dynes). 

7.  Air  is  14.44  times  as  heavy  as  hydrogen.     Find  the  velocity 
of  the  hydrogen  molecule.     What  relations  do  the  densities  of 
gases  bear  to  their  molecular  velocities? 


XVII 

LAW     OF     AVOGADRO— DENSITIES    OF     GASES    AND 

VAPORS 

Theory 

The  speculations  of  the  old  Greek  philosophers  led  some  of 

them  to  the  assumption  of  the  atomic  theory  as  to  the  constitution 

of  matter;    but  it  was  not  until  1803  that  the  English 

Origin  of  the  .          '  . 

atomic  chemist,   John  Dalton,   placed   this   theory   upon   an 

hypothesis.  .     '  '  i  i   *• 

experimental  rather  than  upon  a  purely  speculative 
foundation.  Experiments  upon  the  combining  powers  by  weight 
of  different  substances  reveal  four  principles  which  find  their  most 
simple  and  natural  interpretation  in  the  atomic  hypothesis. 
These  are  (1)  the  principle  of  constant  proportions,  (2)  the  prin- 
ciple of  equivalence,  (3)  the  principle  of  multiple  proportions, 
(4)  the  principle  of  substitution. 

According  to  the  first  principle,  the  proportions  by  weight  in 
which  the   elements  enter  into  a  given   compound  are   absolutely 

invariable.     For  example,  it  is  found  that  hydrogen 

The  law  of  .... 

constant  pro-  will  unite  with  chlorine,  so  as  to  leave  no  free  hydrogen 
and  no  free  chlorine,  only  when  the  number  of  grams 
of  hydrogen  present  bears  to  the  number  of  grams  of  chlorine  one 
definite  ratio.  The  same  may  be  said  of  the  combination  of 
hydrogen  with  bromine  or  of  hydrogen  with  iodine.  These 
ratios  are 

1  gm.  hydrogen  to    35.18  gm.  chlorine  j 

1  gm.  hydrogen  to    79.36  gm.  bromine  >  •         (138) 

1  gm.  hydrogen  to  125.90  gm.  iodine      ) 

Similarly,  the  chlorides  of  potassium,  sodium,  and  silver  are 
always  found  to  contain  exactly  the  following  proportions  by 
weight : 

38.86  gm.  potassium  to  35.18  gm.  chlorine  \ 
22.88  gm.  sodium  to       35.18  gm.  chlorine  >  •    (139) 
107.12  gm.  silver  to          35.18  gm.  chlorine  ) 
138 


DENSITIES    OF    GASES    AXD    VAPORS  139 

It  is  evident  that  the  interpretation  of  this  law  in  the  light  of  a 
molecular  hypothesis  as  to  the  constitution  of  matter  can  only  be 
that  the  atoms  of  each  substance  are  of  constant  weight  and  that 
the  molecules  of  compounds  are  always  of  the  same  atomic  com- 
position. 

But  another  and  still  more  significant  relation  is  found  to 
exist.     From  the  examples  given  above,  it  is  evident  that  35.18 

gm.  of  chlorine,  79.36  gm.  of  bromine,  and  125.90  gm. 

of  iodine  may  be  called  equivalent  quantities  in  the 


sense  that  each  one  of  them  combines  with  exactly  the 
same  weight  of  hydrogen,  viz.  1  gm.  Similarly,  38.86  gm.  of 
potassium,  22.88  gm.  of  sodium,  and  107.12  gm.  of  silver  may 
also  be  called  equivalents,  since  each  of  them  combines  with  the 
same  quantity  of  chlorine.  These  latter  substances  can  not  be 
made  to  combine  with  hydrogen  directly,  but  since  the  numbers 
given  combine  with  just  the  number  of  grams  of  chlorine  which 
has  been  found  to  be  the  combining  equivalent  of  1  gm.  of 
hydrogen,  these  numbars  may  also  be  said  to  have  b3en  found  in 
this  indirect  way  to  be  the  equivalents  in  combining  power  of  1 
gm.  of  hydrogen.  So  much  for  the  definition  of  equivalent. 
Xow  the  fact  of  peculiar  significance  is  this:  A  quantitative 
analysis  of  the  bromides  of  potassium,  sodium,  and  silver  leads  to 
precisely  the  same  numbers  for  the  equivalent  weights  of  these 
substances  as  did  a  study  of  the  chlorides.  Thus  the  only  pro- 
portions in  which  these  substances  will  combine  with  bromine  are: 
38.86  gm.  potassium  to  79.36  gm.  bromine  ) 
22.88  gm.  sodium  to  79.36  gm.  bromine  >••  (140) 
107.12  gm.  silver  to  79.36  gm.  bromine  ) 

When,  further,  a  study  of  the  iodides  leads  again  to  the  same 
three  numbers,  thus, 

38.86  gm.  potassium  combines  with  125.90  gm.  iodine  \ 
22.88  gm.  sodium  combines  with       125.90  gm.  iodine  >  ?  (141) 
107.12  gm.  silver  combines  with          125.90  gm0  iodine  ) 

it  becomes  certain  that  some  very  definite  physical  significance 
lies  behind  these  numbers.  The  simplest  possible  interpretation 
to  put  upon  them  is,  to  take  a  particular  case,  that  the  particle  of 
potassium  which  combines  with  chlorine  to  form  the  molecule  of 
potassium  chloride  is  exactly  liko  the  particle  which  combines  with 


140  MOLECULAR    PHYSICS    AND    HEAT 

bromine  to  form  the  molecule  of  potassium  bromide,  and  with  iodine 
to  form  the  molecule  of  potassium  iodide. 

If,  now,  it  is  decided  to  adopt  a  mere  convention  and  call 
this  particle  an  atom  of  potassium;  if  similarly  the  particle  of 
bromine  which  enters  into  hydrobromic  acid,  and  into  the  bromides 
of  potassium,  sodium,  and  silver  be  called  an  atom  of  bromine; 
and  if  a  similar  convention  be  adopted  with  reference  to  all  of  the 
substances  thus  far  mentioned,  then  the  weights  of  all  these  atoms 
in  terms  of  the  atom  of  hydrogen  must  be  simply  the  numbers, 
above  given,  which  represent  the  combining  powers  with  reference 
to  hydrogen,  of  the  elements  mentioned.  Further,  since  it  has 
been  decided  to  regard  this  quantity  of  each  element  which  enters 
into  the  molecule  of  any  of  the  above  compounds  as  a  unit  rather 
than  as  a  combination  of  two  or  more  units,  the  following  symbols 
will  henceforth  be  used:  hydrochloric  acid  =  HC1,  hydrobromic 
acid  =  HBr,  hydroiodic  acid  =  HI,  potassium  chloride  =  KC1, 
sodium  chloride  =  NaCl,  silver  chloride  =  AgCl.  Corresponding 
formulae  for  the  bromides  and  iodides  are:  KBr,  NaBr,  AgBr, 
KI,  Nal,  Agl.  Of  course,  these  particles  which  enter  into  the 
above  combinations  may  themselves  be  aggregations  of  2,  3,  10,  or 
1000  smaller  particles  for  aught  we  know,  but  so  long  as  no  evi- 
dence is  brought  forward  to  show  that  some  sort  of  compound 
substance  exists,  the  molecule  of  which  contains  a  smaller  amount 
of  chlorine,  for  example,  than  that  quantity  which  is  found  in 
HC1,  KC1,  XaCl,  and  AgCl,  this  quantity  will  be  called  by  com- 
mon consent,  i.e.,  by  definition,  the  atom  of  chlorine.  An  atom 
would  then  be  defined  as  the  smallest  particle  of  any  element 
which  is  known  to  enter  into  the  molecule  of  any  compound. 

The  facts  of  equivalence  which  have  been  above  presented, 
and  which  constitute  one  of  the  strongest  arguments  for  the 
atomic  hypothesis,  may  be  summarized  thus:  The  study  of  many 
different  compounds  leads  often  to  precisely  the  same  number  as  the 
combining  equivalent  of  a  given  element  with  reference  to  hydrogen. 

But  in  some  cases  the  study  of  different  compounds  leads  to 
different  numbers  for  the  equivalent  of  a  given  element  with  refer- 
ence to  hydrogen.  For  example,  Dalton  found  that 
multiple  olefiant  gas  vielded  upon  decomposition  the  two  ele- 

proportions.  r  •       .1.  ,• 

ments    carbon   and  hydrogen   in   the   proportions   by 
weight,  6  carbon  to  1  hydrogen,  while  marsh  gas  yielded  the  same 


DENSITIES    OF    GASES    AND    VAPORS  141 

two  elements  in  the  ratio  3  carbon  to  1  hydrogen.  Further  study 
of  other  compounds  revealed  the  fact  that  whenever  elements  have 
the  power  of  combining  in  different  proportions,  these  proportions 
always  bear  simple  ratios  to  one  another.  This  is  known  as  the 
law  of  multiple  proportions.  The  self-evident  interpretation  of  the 
law,  in  the  light  of  a  molecular  hypothesis,  is  that  it  is  possible  in 
some  cases  for  two,  or  three,  or  some  other  small  number  of  atoms 
of  a  given  element  to  enter  into  the  constitution  of  a  compound 
molecule.  It  was  probably  the  discovery  of  this  law  of  multiple 
proportions  which  first  convinced  Dalton  of  the  truth  of  the 
atomic  theory.  To  illustrate  it  by  a  further  example,  there  are 
four  different  compounds  of  the  elements  chlorine,  potassium, 
and  oxygen  in  which  the  proportions  by  weight  of  the  three  ele- 
ments are  as  follows : 


(143) 


It  will  be  observed  that  in  all  four  compounds  the  potassium  and 
chlorine  have  exactly  the  same  ratios  as  in  KC1.  The  simplest 
and  most  natural  interpretation  is  that  all  of  the  compounds  con- 
tain an  atom  of  potassium  and  an  atom  of  chlorine.  The  smallest 
amount  of  oxygen  found  in  the  molecule  of  any  of  these  com- 
pounds weighs  then  15.88  times  as  much  as  the  hydrogen  atom. 
If  this  amount  be  called  the  oxygen  atom  (at  least  until  some 
smaller  amount  is  found  to  enter  into  some  other  sort  of  molecule), 
then  the  second  compound  contains  two  atoms  of  oxygen,  the  third 
three,  the  fourth  four,  and  the  formulae  for  the  four  substances 
are  KC10,  KC102,  KC103,  and  KC10,. 

It  will  have  been  already  observed  that  the  fact  of  combination 
in  multiple  proportions  introduces  an  uncertainty  into  the  deter- 
The  rind  u  mina^i°n  °^  the  true  combining  equivalents,  i.e.,  the 
of  substitution,  atomic  weights  of  some  elements.  For  example,  from 
Dalton's  experiments  on  olefiant  gas  and  marsh  gas,  it  might  be 
inferred  that  the  atomic  weight  of  carbon  was  6  and  the  formula 
for  olefiant  gas  CH  and  for  marsh  gas  CH2.  But  3  might  with 
equal  reason  be  taken  as  the  atomic  weight  of  carbon,  and  02H 


CHLORINE 

POTASSIUM 

OXYGEN 

35.18 

38.86 

15.88" 

35.18 

38.86 

31.76 

35.18 

38.86 

47.64 

35.18 

38.86 

63.52> 

142  MOLECULAR    PHYSICS    AND    HEAT 

and  OH  as  the  corresponding  formulae.  The  following  experi- 
ment, however,  makes  it  certain  that  the  molecule  of  marsh  gas 
contains  at  least  four  hydrogen  atoms.  It  is  found  that,  by  suc- 
cessive treatments  with  chlorine,  marsh  gas  can  be  made  to  yield 
five  different  compounds  in  which  hydrogen,  carbon,  and  chlorine 
are  combined  in  the  following  proportions  by  weight: 

HYDROGEN        CHLORINE  CARBON 

4  0  11.91 

3  35.18x1  11.91 

2  35.18x2  11.91 

1  35.18x3  11.91 

0  35.18x4  11.91, 

The  existence  of  this  series  proves  that  the  hydrogen  in  marsh  gas 
is  divisible  into  at  least  four  parts,  and  that  1,  2,  3,  or  4  of  these 
parts  may  at  will  be  replaced  by  1,  2,  3,  or  4  atoms  of  chlorine. 
It  is  certain,  then,  that  there  must  be  as  many  as  four  atoms  of 
hydrogen  in  the  molecule  of  marsh  gas.  If  the  quantity  of  carbon 
present  in  this  molecule  be  provisionally  assumed  to  be  the  atom 
(an  assumption  which  is  justified  by  the  study  of  the  other  carbon 
compounds),  the  formula  for  marsh  gas  becomes  CH4;  for  olefiant 
gas,  CH2;  and  the  hydrogen  equivalent  of  carbon,  i.e.,  its 
atomic  weight,  is  fixed  not  at  6  or  3  but  at  12  (accurately  at 
11.91). 

Again,  since  water  is  found  to  contain  hydrogen  and  oxygen  in 
the  ratio  1  to  7.94,  this  might  at  first  be  taken  as  the  ratio  of  the 
weights  of  the  atoms  of  hydrogen  and  oxygen,  and  the  symbol  for 
water  written  HO.  This  would  indeed  be  inconsistent  with  the 
interpretation  put  upon  the  series  (142),  from  which  it  was 
inferred  that  the  atomic  weight  of  oxygen  was  15.88.  It  would 
be  possible,  however,  to  reconcile  this  difficulty  by  assuming  the 
formulae  for  the  compounds  in  (142)  to  be  KC102,  KC104, 
KC106,  KC108, — an  assumption  not  very  natural,  it  is  true,  since 
the  new  formulae  at  once  suggest  that  it  is  02  rather  than  0, 
which  is  the  oxygen  unit.  But  the  principle  of  substitution  leaves 
no  doubt  as  to  which  of  the  above  possibilities  must  be  chosen. 
For  if  potassium  be  allowed  to  act  on  water,  a  portion  of  the 
hydrogen  is  drawn  off  and  replaced  by  potassium ;  if  it  be  allowed 
to  act  again,  all  of  the  hydrogen  is  replaced,  the  proportions  by 
weight  in  the  three  compounds  being  as  follows: 


DENSITIES    OF    GASES    AND    VAPORS  143 


HYDROGEN 

POTASSIUM 

OXYGEN 

2 

0 

15.88 

1 

38..S6 

15.88 

0 

77.72 

15.88 

This  series  makes  it  certain  that  there  are  at  least  two  atoms  of 

hydrogen  in  the  water  molecule,  and  therefore  supports  the  first 

conclusion  that  the  atomic  weight  of  oxygen  is  15.88,  and  the 

constitution  of  water  H20.      This  gradual  replacement  of  a  given 

element  of  a  compound  by  successive  reactions  is  called  substitution. 

After  the  atomic  weights  of  carbon  and  oxygen  have  been  fixed 

at  11.91  and  15.88,  the  discovery  that  carbon  monoxide  contains 

these  elements  in  exactly  these  proportions,  and  that 

Some  other  J  .  ' 

atomic  carbon  dioxide  contains  the  same  elements  in  the  pro- 

cjie'micni  portions  11.91  to  31.76,  leads  at  once  to  the  formulae 
CO  and  C02  to  represent  the  chemical  constituents  of 
these  gases.  Again,  when  carbon  and  nitrogen  are  found  to 
unite  in  the  ratio  11.91  to  13.93,  and  carbon,  nitrogen,  and 
hydrogen  in  the  ratio 

CARBON  NITROGEN  HYDROGEN 

11.91       :       13.93         :         1 

it  is  natural  to  take  13.93  as  the  probable  value  of  the  weight 
of  the  nitrogen  atom.  Then  the  formula  for  nitric  oxide,  which 
contains  13.93  gm.  of  nitrogen  to  15.88  gm.  of  oxygen,  becomes 
NO,  and  that  for  nitrous  oxide,  in  which  nitrogen  and  oxygen  are 
found  in  the  ratio  13.93  to  7,94,  becomes  N20.  Furthermore, 
the  fact  that  sulphur  combines  with  chlorine  in  the  ratio  31.83 
to  35.18,  and  also  with  bromine  in  the  ratio  31.83  to  79.36,  sug- 
gests 31.83  as  the  atomic  weight  of  sulphur. 

Enough  has  now  been  said  to  show  how,  aided  only  by  the 
laws  of  combination,  chemists,  beginning  with  Dalton,  set  about 

devising  tables  of  atomic  weights  and  molecular  consti- 
Awgadro  tutions;  tables  which,  to  be  sure,  were  only  provisional, 

since  it  might  sometimes  be  difficult  to  determine 
whether  the  atomic  weight  of  a  substance  should  be  represented 
by  a  certain  number,  or  by  some  simple  multiple  or  sub-multiple 
of  that  number.  But  as  more  compounds  were  investigated,  the 
choices  became  more  and  more  restricted,  and  it  can  scarcely  be 
doubted  that  the  study  of  combining  powers  alone  would  have  led 


144  MOLECULAR    PHYSICS    AND    HEAT 

ultimately  to  most  of  the  now  accepted  values  of  atomic  weights 
and  chemical  formulae,  even  if  Avogadro's  Law  had  never  been 
discovered.  The  discovery  of  this  law,  however,  facilitated 
greatly  the  work  of  fixing  these  quantities.  The  law  was 
announced  in  1811  by  the  Italian  chemist  whose  name  it  bears. 
It  asserts  that  at  a  given  temperature  and  pressure  all  gases  contain 
the  same  number  of  molecules  per  cubic  centimeter.  The  proof  of 
the  law  rests  upon  a  remarkable  relation  which  is  found  to  exist 
between  the  combining  powers  of  substances  and  their  gas  or 
vapor  densities.  The  following  table*  brings  out  clearly  this 
striking  relation.  The  column  headed  "Density"  represents  the 
results  of  experiments  upon  the  relative  densities,  at  a  given 
temperature  and  pressure,  of  a  number  of  the  gases  already  men- 
tioned. For  convenience  of  representation,  the  density  of  hydro- 
gen gas  is  taken  as  2. 

GAS  DENSITY  ATOMIC   WEIGHT 

Hydrogen 2  1 

Nitrogen 27.82  13.93 

Oxygen 31.80  15.88 

Chlorine  70.72  35.18 

Bromine 159.54  79.36 

Iodine 254.73  125.90 

Sulphur 64.06     '          31.83 

MOLECULAR  WEIGHT 

Hydrochloric  acid  (HC1) 36.30  36.18  U  +  35.18) 

Marsh  gas  (CH4) 16.08  15.91  (11.91  + 4) 

Carbon  monoxide  (CO) 27.95  27.79  (11.91  +  15.88) 

Carbon  dioxide  (C02) 44.10  43.67  (ii.Qi  + 15.88  x  2) 

Nitric  oxide  (NO) 29.95  29.81  (13.93  +  15.88) 

Nitrous  oxide  (N20) 44.10  43.74  (13.93  x  2  +  15.88) 

Water  (H30) 18.03  17.88  (2+15.88) 

It  is  seen  that  in  the  lower  group  of  substances  the  numbers 
which  represent  vapor  densities  in  terms  of  a  gas  one-half  as  dense 
as  hydrogen  are  throughout  almost  identical  with  the  numbers 
which  represent  the  weights  of  the  molecule  in  terms  of  the  weight 
of  the  hydrogen  atom,  as  above  defined.  But  if  the  weights  of 
the  individual  molecules  of  a  number  of  gases  bear  the  same  ratios 

*See  Landolt  and  Bornstein,  Physikalisch-chemische  Tabellen,  pp. 
115,  116;  and  Wullner,  Experimental  Physik,  Vol.  II,  p.  802. 


DENSITIES   OF    GASES    AND    VAPORS  145 

as  the  weights  of  the  gases  per  cc.,  then  evidently  the  number  of 
molecules  per  cc.  must  be  the  same  in  all  the  gases.  This 
remarkably  simple  conclusion,  which  applies  necessarily  to  all  of 
the  gases  of  the  second  group,  provided  the  conclusions  above 
reached  as  to  their  molecular  constituents  are  correct,  is  seen  to 
apply  also  to  all  the  gases  of  the  first  group,  if  only  the  molecules 
of  the  gases,  hydrogen,  nitrogen,  oxygen,  chlorine,  bromine, 
iodine,  and  sulphur,  be  assumed  to  be  twice  as  heavy  as  the  atoms  . 
of  these  substances  ;  that  is,  if  these  molecules  are  composed  each 
of  two  atoms,  thus,  H2,  X2,  02,  C12,  Br2,  I2,  S2;  for  then  the 
molecular  weights  become  2,  27.86,  31.76,  70.36,  158.72,  251.80, 
arid  63.66  respectively,  numbers  which  are  in  remarkably  close 
agreement  with  those  given  in  the  column  of  densities. 

Xow,  it  is  found  that  with  equally  simple  choices  as  to  the 
molecular  constitutions  of  those  gases  in  which  thajcombining 
powers  of  the  constituent  elements  leave  two  or  m6ra  choices 
open,  the  densities  of  all  known  gases  become  identicg]Lwith  their 
molecular  weights.  This  constitutes  overwhelmm^evidence  for 


the  truth  of  Avogadro's  hypothesis.  It  is  this  fa$  of  agreement 
between  molecular  weights  and  vapor  densities  v&y&Js  the  experi- 
mental basis  for  the  law  of  Avoyadro.*  Tliis  agreement  is  least 


perfect  in  the  cases  of  those  gases  which  Sh  the  largest  depar- 
tures from  Boyle's  Law.  For  actual  ga^es  this  law,  therefore, 
like  those  of  Boyle,and  Gay-Lussac,  is  only  a  close  approximation. 

-Experiment 

1.  T<^  ue^mine  the  density  of  C02  and  to  compare 
the  same  wjth  its  molecular  weight. 

2.  To  determine,*,,the  density  ot^Vfater  vapor  and  to  compare 
the  same  with  its.  molecular  weight.^  * 

A  glass  globe^bf  known  volume  V  is  weighed,  first  when  full  of 
air  at    temperature   T±    (absolute),   pressure  Px,   then  when  full 
of  the  unknowi^ts  at  temperature  T2,  pressure  Pt. 


In  these  weighings  a*closed  bulb  of  the  same  volume  as 

*The  proof  of  this  law*  which  Maxwell  first  drew  from  the  kinetic 
theory — a  proof  which  rests  upon  the  Maxwell-Boltzman  law  of  the  dis- 
tribution of  energies  between  two  sets  of  unlike  particles  in  a  gaseous 
mixture,  and  which  has  since  been  given  a  place  in  a  large  number  of 
chemical  and  physical  texts — can  not  be  recognized  as  adequate.  (See 
Note  by  Rayleigh  in  Maxwell's  Theory  of  Heat,  10th  ed.,  p.  326.) 


140  MOLECULAR    PHYSICS    AND    HEAT 

the  density  globe  is  used  as  a  counterpoise  so  as  to  eliminate  all 
effects  due  to  changes  in  the  buoyancy  of  the  air.  If  the  differ- 
ence between  the  first  and  second  weighings  be  represented  by  w 
(w  being  of  course  negative  if  the  second  weight  exceeds  the 
first),  the  density  of  the  gas  at  TZPZ  by  d02,  the  density  of  air 
at  T^  by  dai,  then  evidently 

Vdai-  Vdn=w*  (145) 

Further,  since  density  is  directly  proportional  to  pressure  when 
the  temperature  remains  constant  (see  Boyle's  Law),  and  inversely 
proportional  to  absolute  temperature  when  the  pressure  remains 
constant  (see  Ex.  XVI,  Problem  2)  ,  it  is  evident  that  the  equation 
which  expresses  the  relation  between  the  densities  of  air  at  TlPl 
and  at  T2P2  is 


d     ~  p 

#a2         r<*     2  1 

Now,  the  quantity  which  will  be  first  sought  in  this  experiment  is 
the  density  of  the  unknown  gas  in  terms  of  the  density  of  air  at 

the  same  temperature  and   pressure,  i.e.,  -p--    This   is   obtained 

dai 

easily  from   (145)  and  (146).     Thus,  substitution  in  (145)  of  the 
value  of  dai  obtained  from  (146)  gives 


TTTl  ai)  i  Z  W 

whence  f:2=psTrv^- 

All  of  the  quantities  on  the  right  side  of  (147),  excepting^, 
are  measured  directly  in  the  experiment.  da2  is  obtained  from 
(146)  and  the  result  of  Ex.  XV;  or,  if  it  is  found  more  convenient 
to  determine  dai  than  da^  (147)  may,  with  the  aid  of  (146),  be 
thrown  into  the  form 


*In  this  equation  the  expansion  of  the  bulb  is  neglected,  because  in 
neither  of  the  following  determinations  will  it  affect  the  result  by  more 
than  a  small  fraction  of  one  per  cent.  If  it  is  desired  to  take  it  into 
account  it  is  only  necessary  to  replace  (145)  by  Vdai  —  V(l  -}-  yt)daa  =  w 
(in  which  7  is  the  expansion  coefficient  of  glass  and  t  the  number  of 
degrees  between  Tl  and  T2),  and  then  to  solve  precisely  as  above. 


DENSITIES    OF    GASES    AND    VAPORS  147 

Finally,  since  air  is  14.44  times  heavier  than  hydrogen,  it  is  only 
necessary  to  multiply  the  density  of  the  unknown  gas  in  terms  of 
air  by  28.88  in  order  to  obtain  its  density  in  terms  of  a  gas  one- 
half  as  heavy  as  hydrogen.  This  is  the  quantity  which,  accord- 
ing to  the  law  of  Avogadro,  should  agree  with  the  molecular 
weight. 

DIRECTIONS. — 1.  For  convenience  in  filling  and  weighing,  the 

density  globe,  the  capacity  of  which  is  about  250  cc.,  is  provided 

with  two  taps  a  and  b  (see  Fig.  80).     The  volume  of 

this  globe  is  first  to  be  found  by  filling  it  with  water 

and  weighing  upon  the  trip  scales.      The  density  of   water  at  the 

observed  temperature  is  to  be  taken  from  the  table  of   water 


FIGURE  80 

densities    in  the  Appendix.      V  is,  of    course,  the  ratio  of  the 
weight  and  density  of  the  water. 

Having  found  F,  dry  the  globe  by  carefully  rinsing  it  with 
alcohol  and  then  forcing  through  it  a  current  of  air  from  a  bel- 
lows, at  the  same  time  heating  srentlv  bv  means  of  a 

Filling  with  »  °         r      J     L.  . 

air  and  rapidly  moving  Bunsen  flame.  Continue  this  opera- 
tion until  all  odor  of  alcohol  has  disappeared  from  the 
bulb.  Then,  after  carefully  lubricating  the  stopcocks,  connect 
the  bulb  with  the  bellows  through  a  calcium  chloride  drying- 
tube,  as  in  Fig.  80,  and  force  through  the  combination  a  very 
gentle  current  of  air  for  about  one  minute.  Then  close  tap  b  and 
allow  the  apparatus  to  stand  in  this  condition  for  about  five  min- 
utes, shielding  the  bulb  as  much  as  possible  from  temperature 
changes.  Xext  close  tap  «,  read  the  barometer,  and  take  the 
temperature  by  means  of  a  thermometer  hung  near  the  bulb. 
Then  detach  the  bulb,  carefully  remove  all  dust  and  grease  from 


148  MOLECULAR    PHYSICS    AND    HEAT 

its  surface,  and  weigh  upon  an  analytical  balance,  using  a  counter- 
poise of  the  same  volume  as  the  globe  and  following  the  directions 
given  in  Ex.  XV  for  the  first  weighing.  It  is  particularly  impor- 
tant that  no  mercury  be  allowed  to  touch  the  bulb,  as  it  is  nearly 
impossible  to  remove  small  mercury  globules  from  a  glass  sur- 
face. 

After  weighing,  put  the  bulb  into  connection,  through  the 
drying-tube,  with  a  reservoir. of  C02  or  with  a  vessel  in  which  the 

gas  is  being  generated;  and,  keeping  the  bulb  in  such 
co,  and  position  that  the  exit  tap  is  on  top,  allow  a  gentle  current 

of  the  gas  to  flow  through  the  bulb  for  about  two  min- 
utes. Find  the  temperature  of  the  issuing  gas  by  placing,  at  the 
orifice  of  the  exit  tap,  the  bulb  of  the  thermometer  previously  used. 
Then  close  first  the  entrance,  then  the  exit  tap ;  detach  the  bulb 
and  re-weigh,  following  the  directions  given  in  Ex.  XV  for  the  sec- 
ond weighing.  If  neither  the  temperature  nor  the  pressure  differs 
appreciably  from  the  values  taken  in  connection  with  the  first 
weighing,  (147)  reduces  to 

rfw-i " 

da,~     ~Vda,' 

w  being  itself  negative  in  this  case,  since  C02  is  heavier  than  air.* 

2.  The  method  here  used  for  determining  the  density  of  water 

vapor  in  terms    of  air  does  not  differ  in  principle   from   that 

employed  with  C02.  Since,  however,  the  maximum 
Filling  the  pressure  which  water  vapor  is  able  to  exert  at  ordinary 

temperatures  is  less  than  atmospheric  pressure  (see  Ex. 

XVIII),  it  is  evident  that  it  must  be  impossible  under 
ordinary  atmospheric  conditions  wholly  to  replace  the  air  in  the 
density  globe  by  water  vapor.  This  can  be  done  easily  at  any  tem- 
perature at  which  the  maximum  pressure  of  water  vapor  is  more 
than  atmospheric,  e.g.,  at  150°C.  Hence  the  following  direc- 
tions :  After  carefully  drying  the  bulb  B  (see  Fig.  81),  by  repeatedly 

*Precisely  the  same  method  may  be  used  with  gases  lighter  than  air, 
save  that  in  this  case  the  exit  tap  should  be  at  the  bottom  during  the 
filling.  However,  all  of  the  standard  determinations  of  gas  densities  have 
been  made  with  bulbs  provided  with  but  one  tap  rather  than  with  two  as 
here  described.  The  bulb  has  then  been  completely  exhausted  before 
being  put  into  connection  with  the  gas  reservoir. 


DENSITIES    OF    GASES    AND    VAPORS 


149 


warming  and  exhausting  through  a  calcium  chloride  tube  as  in 
Ex.  XV,  make  a  first  weighing  upon  an  analytical  balance,  using, 
as  above,  a  counterpoise  of  the 
same  volume  as  B.  Then  place 
the  capillary  orifice  o  beneath 
the  surface  of  distilled  water, 
and  warm  the  bulb  slightly  by 
means  of  the  hand.  Upon 
cooling,  one  or  two  cubic  cen- 
timeters of  water  will  be  drawn 
into  the  bulb.  Next,  com- 
pletely immerse  the  bulb  in 
melted  paraffin  (see  Fig.  81), 
leaving  but  a  few  centimeters 
of  the  capillary  tube  projecting 
above  the  surface  of  the  liquid. 
Keep  the  temperature  of  the 
bath  constant,  at,  say,  120° 
C.,  by  very  thorough  and  con- 
tinuous stirring,  and  by  a 
proper  regulation  of  the  Bun- 
sen  burner.  The  rapid  vapor- 
ization of  the  water  will  drive  FIGURE  si 
all  air  from  B.  If  a  flame 

"be  held  in  front  of  0,  it  will  be  seen  to  be  deflected  by  the  rapid 
current  of  issuing  steam.  When  the  cessation  of  this  deflec- 
tion indicates  that  the  water  in  B  has  entirely  boiled  a\vay,  seal 
the  globe  by  means  of  a  fine  blow-pipe  flame.  This  is  best  don£ 
by  heating  the  tube  to  softness  just  above  the  paraffin  and  then 
drawing  off  the  tip.  As  soon  as  the  sealing  is  complete,  take 
readings  of  the  temperature  of  the  bath  and  of  the  barometer 
height.  Then  remove  the  bulb,  clean  it  thoroughly  with  a  cloth 
while  the  paraffin  is  still  hot,  and  test  for  a  leak  by  allowing  the 
condensed  steam  to  run  down  to  the  tip  of  the  tube  and  observing 
whether  or  not  fine  bubbles  enter  the  bulb.  Then,  after  cooling 
to  the  temperature  of  the  room,  again  weigh  the  bulb  together 
with  the  drawn-off  tip. 

To  find  the  volume  of  the  bulb,  file  off  under  water  the  sealed 
tip.     The  bulb  will  at  once  fill  with  water.     Weigh  this  bulb  full 


150  MOLECULAR    PHYSICS    AND    HEAT 

of  water  upon  the  trip  scales,  and  compute  the  volume  as  in  the 
experiment  with  C02.  If  the  bulb  does  not  completely  fill  with 

water,  the  filling  may  be  completed  by  means  of  a 
buibmeof  Pipette.  It  is  true  that  (147)  and  (148)  are  not  then 

rigorously  correct,  but  unless  the  bubble  is  quite  large, 
the  error  introduced  will  be  negligible. 


Record 

1.  Weight  on  trip  scales  of  bulb  =  —          -  of  bulb  -f  water  =  — 
Temperature  of  water  =  —  .  •.  density  of  water  =  —          .  •.  V  = 


Wt.  added  to  counterpoise  to  balance  globe  -f-  air  = rest'g  pt.  =  — 

Wt.  added  to  counterpoise  to  balance  globe  -f-  CO2  = rest'g  pt.  =  — 

Rest'g  pt.  after  addition  of  2  mg.= .-.  sensitiveness  = .\w  =  — 

•'•fa-=  —      -X  28.88  =  -      -  molecular  wt.  =—       -  %  error  =  - 

Wt.  added  to  counterpoise  to  balance  globe  -f-  air  = rest'gpt.  =  — 

Wt.  added  to  counterpoise  to  balance  globe  -f-  H2O  = rest'g  pt.  = 

Rest'g  pt.  after  addition  of  2  mg.  = .•.  sensitiveness  = .•.«?  =  — 

Weight  on  trip  scales  of  bulb  =  —                      of  bulb  -f  water  =  — 
Temperature  of  water  = .  •.  density  of  water  = .:  V= 


X  28.88  =  --  molecular  wt.  = 


error  =  -- 


Problems 

1.  One  gram  of  air  was  introduced  into  an  empty  spherical 
bulb  of  radius  10  cm.     Find  the  pressure,  in  mm.  of  mercury, 
which  the  gas  exerted  against  the  walls  of  the  bulb  when  the 
temperature  was  25 °C. 

2.  Find  the  pressure  which  the  same  weight  of  N^O  gas  would 
exert  in  the  same  vessel  at    the  same  temperature.      Compute 
similarly  for  hydrogen  gas;  for  CH4  gas. 

3.  One  gram  of  nitrogen  and  1  gram  of  HJS  are  introduced 
together  into  the  bulb  used  in  the  first  Problem.     Find  the  pres- 
sure at  25 °C 


DENSITIES    OF    GASES    AND    VAPORS  151 

4.  Find  the  volume  which  2  grams  of  hydrogen  will  occupy  at 
0°C.  76  cm.     The  same  for  32  grams  of  oxygen;  for  30  grams 
NO.     Explain  the  connection  between  the  results. 

5.  Find  the  density  of  air  at  100 °C.  76  cm. ;    of  water  vapor. 
If   the  density  of    water  at    100°C.  76  cm.  is  .95852,  find  how 
many  times  water  expands  upon  vaporizing. 

6.  Find  the  density  of  alcohol  vapor  at  72°  76  cm.,  the  formula 
for  alcohol  being  C2H60.     If  the  density  of  alcohol  at  72°  is  .8, 
find  how  many  times  alcohol  expands  upon  vaporizing. 


XVIII 

THE  PRESSURE  -  TEMPERATURE   CURVE  OF  A 
SATURATED  VAPOR 

Theory 

If  the  molecules  of  gases  and  of  vapors  are  in  rapid  motion,  the 
molecules  of    liquids  must  be  also,  for  no  fundamental  distinc- 
tion exists  between  the  liquid  and  the  gaseous  con,di- 
theoryof        tions.     At  high  temperatures  the  two  states  become 
absolutely  identical.      At   temperatures  below  a   cer- 
tain critical  value,  however,   the  possession  of  a  clearly  marked 
surface  may  be  taken  as  the  distinguishing  feature  of  a  liquid. 

Figures  82  and  83  illustrate  the  probable  differences  between 
the  motions  of  the  molecules  in  gases  and  in  liquids  at  ordinary 
temperatures  and  pressures.  In  the  former  (Fig.  82)  the  mole- 


FlGURE  82 

cules  are  so  far  apart  that  their  mutual  attractions  may  in  general 
be  neglected.  They  move  in  straight  lines  through  distances 
which  are  large  in  comparison  with  their  own  dimensions.  Their 
motions  change  direction  at  collision  only.  The  zigzag  line  repre- 
sents a  possible  path  of  one  single  molecule  in  going  from  position  1 
to  position  1'  and  making  impacts  in  so  doing  against  molecules 
2,  3,  4,  etc.,  up  to  12.  The  mean  distances  traversed  between 

152 


TENSION    OF   SATURATED    VAPOR  153 

successive  collisions  by  a  molecule  of  air  at  0°C.,  76  cm.,  is  only 
about  .00006  mm.,  but  this  is  a  distance  which,  small  as  it  is,  is 
at  least  100  times  as  large  as  the  diameter  of  the  molecule.  In 
liquids,  on  the  other  hand  (see  Fig.  83),  the  molecules  are  crowded 
so  closely  together  that  their  mo- 
tions between  impacts  are  extremely 
minute — of  the  same  order  of  mag- 
nitude as  the  molecules  themselves 
— and  at  the  surface  of  the  liquid, 
where  there  is  greater  freedom  of 
motion,  the  paths  of  the  particles 
are  influenced  not  only  by  collisions 
but  also  by  the  attractions  of  the 
other  molecules.  On  account  of  FIGURE  83 

the  enormous  number  of  molecules 

present  in  or  near  the  surface,  this  downward  force  upon  a  mole- 
cule just  above  the  surface  is  doubtless  very  large;  so  large,  in 
fact,  that  the  molecules  which  are  moving  away  from  the  surface 
are  in  general  unable  to  leave  it.  They  simply  rise  to  a  certain 
distance  by  virtue  of  their  velocities,  after  the  manner  of  project- 
iles shot  up  from  the  earth,  and  then  fall  back  again  into  the 
liquid.  Their  paths  near  the  surface  thus  become  similar  to  the 
forms  shown  in  Fig.  83.  The  zigzag  line  in  the  figure  repre- 
sents a  possible  path  of  a  particle  in  the  body  of  a  liquid. 

But  conditions  may  arise  which  render  possible  the  escape  of  a 

molecule  from  the  liquid;   for  it  must  not  be  assumed,  either  in 

the  case  of  liquids  or  of  erases,  that  the  molecules  of 

The  kinetic 

theory  and      the  same  substance  all  have  the  same  velocity,  for  even 

vaporization.  £      . 

if  they  were  all  given  a  common  velocity  to  begin  with, 
the  collisions  would  at  once  create  differences.  Again,  although 
constancy  of  temperature  means  that  the  mean  velocity  reniains 
the  same,  the  velocity  of  each  individual  molecule  will  in  general 
change  at  each  impact.  .  The  conditions  of  impact  must  often  be 
such  that  a  molecule  receives  a  velocity  much  greater  than  the 
mean  value.  If  the  substance  be  a  liquid,  and  if  one  of  these 
more  rapidly  moving  molecules  be  near  the  surface,  it  maybe  able 
to  break  through  the  thin  space  in  which  the  powerful  attraction 
exists,  and  to  move  off  as  an  independent  molecule  into  the  space 
above.  It  thus  happens  that  the  space  above  the  liquid  in  a 


154  MOLECULAR    PHYSICS   AND    HEAT 

closed  vessel  gradually  becomes  filled  with  the  gaseous  form  of  the 
substance  of  the  liquid.  This  gas  becomes  more  and  more  dense 
as  more  molecules  escape,  but  there  is  evidently  a  limit  to  its  pos- 
sible density,  for  many  of  the  escaped  molecules  chance,  in  their 
wanderings,  to  return  to  the  surface  and  reenter  the  liquid.  The 
number  of  molecules  thus  returning  per  second  evidently  increases 
as  the  number  of  molecules  above  the  liquid  increases,  i.e.,  it  is 
proportional  to  the  density  of  the  vapor.  When  this  density  has 
reached  a  certain  limit,  there  is  set  up  a  condition  of  "active 
equilibrium"  in  which  as  many  molecules  reenter  the  liquid  per 
second  as  escape.  When  this  condition  is  reached,  the  vapor  is 
.said  to  be  saturated,  that  is,  it  has  the  largest  density  which  it  is 
ever  able  to  have  at  the  existing  temperature,  and  it  therefore 
exerts  the  largest  pressure  which  it  ever  can  exert  at  this  tem- 
perature. 

But,  if  the  temperature  be  raised,  the  vapor  can  evidently  have 

a  larger  density,  for  the  number  of  molecules  escaping  per  second 

must  be  greater  at  the  higher  temperature,  i.e.,  at  the 

Dependence      ...  _  ^  ' 

of  the  den-       higher  mean  velocity,  and  hence  the  density  ot  the 
pressure  of     vapor  must  be  greater  before  the  condition  of  equilib- 

a  saturated         ...  .,          .          ,.  ,     i  -,      ,  i 

vapor  upon     rium  is  set  up.     Also,  since  the  pressure  exerted  by  the 

temperature.  . 

vapor  is  proportional  both  to  the  density  and  to  the 
mean  kinetic  energy  of  each  impact,  and  since  both  density  and 
kinetic  energy  increase  with  temperature,  it  is  evident  that  the 
pressure  must  rise  with  two-fold  rapidity  as  the  temperature  rises. 
But,  if  the  temperature  be  held  constant,  all  attempts  to 
increase  the  density  or  the  pressure  of  a  vapor  which  is  in  con- 
Density  and  ^ac^  w^n  ^s  liquid  in  a  closed  vessel  must  be  futile. 
independent  ^°  see  ^is  clearly,  conceive  of  a  few  drops  of  ether 
of  volume.  inserted  into  a  barometer  so  as  to  fill  the  space  above 
the  mercury  with  ether  and  saturated  ether  vapor  (see  Fig.  84). 
As  soon  as  the  density  of  this  vapor  is  temporarily  increased  by 
compression,  i.e.,  by  lowering  the  tube  in  the  cistern,  the  equi- 
librium at  the  surface  is  destroyed,  and  immediately  more  mole- 
cules begin  to  enter  the  liquid  per  second  than  escape  from  it. 
Hence,  in  a  very  short  time,  enough  ether  condenses  to  restore  the 
old  condition  of  density  and  pressure.  Similarly,  raising  the  tube 
and  thus  increasing  the  volume  occupied  by  the  vapor  diminishes 
the  number  of  molecules  which  reenter  the  liquid  per  second, 


TENSION    OF   SATURATED   VAPOR  155 

while  leaving  the  number  which  escape,  unchanged ;  hence 

equilibrium  is  soon  reestablished  at  the  old  density  and 

pressure.     This   can  be  proved  easily  by  observing  that 

raising  or  lowering  the  tube  does  not  alter  the  height  of 

the  mercury  in  the  tube  above  the  mercury  in  the  cistern. 

This  readjustment  to  equilibrium  conditions  takes  place 

almost  instantly  when  the  space  contains  only  the  liquid 

and  its  vapor.     The  presence  of  air  or  of  any 

Retarding  ,      ,  .    a 

influence  other  gas  exerts  a  very  marked  influence  upon 
the  time  required  for  adjustment,  but  does  not 
affect  the  ultimate  result.  Thus,  in  Fig.  84,  the  density 
and  pressure  of  the  ether  vapor  at  a  given  temperature  are 
ultimately  the  same  whether  air  is  present  in  the  tube  or 
not,  i.e.,  just  as  much  liquid  will  evaporate  into  a  space 
filled  with  air  as  into  a  vacuum.  But  whereas,  when 
the  ether  is  introduced  into  a  vacuum,  the  maximum 
density  of  the  vapor  is  reached  in  a  few  seconds,  when  it 
is  introduced  into  a  space  containing  air,  the  condition  of 
saturation  may  not  be  reached  for  a  number  of  hours. 
Of  course,  if  air  be  present,  the  total  pressure  against  the 
walls  is  the  sum  of  the  pressures  of  the  air  and  of  the 
vapor. 

If  the  liquid  be  contained  in  a  vessel  which  is  open  to  FIGUB 
the  air,  so  that  the  pressure  can  not  rise  above  the  atmospheric 
condition,  the  vapor  which  forms  is  continually  being 

Effect  of  air  '         .  .  ,    ,  J  ? 

currents  upon  removed  by  diffusion  and  by  air  currents,  so  that  it 
never  reaches  its  maximum  density,  i.e.,  the  rate  of 
exit  of  molecules  from  the  liquid  always  remains  greater  than  the 
rate  of  entry.  Hence  the  liquid  gradually  disappears  or  ''evap- 
orates." The  rate  of  this  evaporation  evidently  depends  upon  th 
rapidity  with  which  the  vapor  above  the  liquid  is  removed.  Hence 
it  is  that  fanning  greatly  facilitates  evaporation. 

The  cooling  effect  of  evaporation  is  very  easily  understood  in 

the  light  of  the  kinetic  theory.     For,  since  it  is  only  the  more 

rapidly  moving  molecules  which  are  able  to  break  away 

The  cooling      *  %_  ,.  ,.    ,          •   ,  ,,.  -  ,, 

effect  of          from  the  attraction  which  exists  near  the  surface,  the 

mean  kinetic  energy  of  the  molecules  of  the  liquid  is 

continually  being  diminished  by  the  loss  of  the  most  energetic 

members.     And  since  temperature  is  a  function  of  the  average 


•j 


156  MOLECULAR    PHYSICS    AND   HEAT 

molecular  energy,  this  loss  means,  of  course,  a  continual  reduction 
of  the  temperature  of  the  liquid.  This  fall  of  temperature  would 
continue  indefinitely  if  the  liquid  did  not  soon  become  so  much 
cooler  than  the  surrounding  objects  that  it  receives  heat  from 
them  as  rapidly  as  it  loses  it  by  evaporation. 

This  passage  from  the  liquid  to  the  vaporous  condition  by  sur- 
face evaporation  takes  place  to  some  extent  at  all  temperatures 
The  toning  wnenever  tne  space  above  the  liquid  is  not'  saturated. 
temperature.  AS  the  temperature  is  increased,  outside  conditions 
remaining  the  same,  it  takes  place  more  and  more  rapidly,  until 
finally  a  temperature  is  reached  at  which  it  begins  to.  take  place 
not  simply  at  the  surface  but  also  within  the  body  of  the  liquid, 
i.e.,  bubbles  of  vapor  begin  to  form  beneath  the  surface  upon  the 
sides  of  the  containing  vessel,  whence  they  rise  to  the  top,  grow- 
ing rapidly  as  they  ascend.  It  is  at  once  evident  that  this  condi- 
tion can  not  be  reached  until  the  maximum  pressure  exerted  by 
the  vapor  which  is  formed  from  the  liquid  is  at  least  equal  to  the 
outside  pressure;  for,  if  the  pressure  exerted  by  the  vapor  in  the 
bubbles  were  less  than  that  outside,  these  bubbles,  even  if  formed, 
would  at  once  collapse.,  This  temperature,  then,  at  which  the 
pressure  of  the  saturated  vapor  becomes  equal  to  the  outside  pressure, 
is  called  the  boiling  temperature.  It  does  not  follow,  however,  that 
a  liquid  will  always  boil  as  soon  as  its  temperature  reaches  the 
boiling  point  as  here  defined.  In  fact,  the  temperature  of  a 
boiling  liquid  must  always  be  at  least  a  trifle  higher  than  that  at 
which  the  pressure  of  the  saturated  vapor  is  equal  to  the  outside 
pressure,  for  the  pressure  within  the  bubble  must  be  sufficient  to 
overcome  not  only  the  outside  pressure  but  also  the  weight  of  the 
superimposed  liquid  and  the  surface  tension  of  the  bubble  (see 
Ex.  XXI).  When  the  bubble,  however,  rises  to  the  surface  and 
breaks,  the  pressure  exerted  by  the  vapor  which  was  contained 
within  it  must  fall  exactly  to  the  atmospheric  condition,  and  the 
temperature  of  this  vapor  must  also  fall,  by  virtue  of  expansion, 
to  that  temperature  at  which  the  pressura  of  th^  saturated  vapor 
is  equal  to  the  existing  atmospheric  pressure.  Hence,  a  ther- 
mometer ivhich  is  to  indicate  the  true  boiling  temperature  according 
to  the  above  definition  must  be  placed  not  in  the  boiling  liquid  itself 
but  in  the  vapor  rising  from  it. 

The  temperature  of  the  liquid  itself  is  a  very  uncertain  quan- 


TENSION    OF    SATURATED    VAPOR  157 

tity.     Gay-Lussac  found  that  the  temperature  of  boiling  water  in 
a  glass  vessel  was  usually  1°  to  3°  higher  than  in  a  metal  ves- 
sel.    For  the  reasons  above  mentioned,  it  must  always  be 

The  temper-          ,    .«     i  .    ,         ,-,          .,       .     ...  •    ,      ,  -, 

ature  of         a  trine  higher  than  the  boiling;  point ;  but  under  some 

theliquid.  . 

circumstances  it  may  rise  many  degrees  above  this 
temperature.  For  it  is  by  no  means  necessary  that  bubbles  of 
vapor  begin  to  form  as  soon  as  the  temperature  is  reached  at  which 
they  are  able  to  exist  after  being  formed.  The  presence  of  air  in 
the  water  or  occluded  in  the  walls  of  the  containing  vessel  is 
found  to  be  essential  to  the  genesis  of  bubbles.  A  Frenchman 
named  Donny  found,  in  1846,  that  when  he  very  carefully  removed 
this  air  he  could  raise  the  temperature  of  water  in  a  glass  vessel  to 
138 °C.  before  boiling  began.  But  in  all  such  cases,  since  the 
pressure  of  the  saturated  vapor  corresponding  to  the  temperature 
of  the  water  is  much  more  than  the  atmospheric  pressure,  as  soon 
as  a  bubble  once  starts  it  grows  with  explosive  rapidity  and 
produces  the  familiar  phenomenon  of  "boiling  with  bumping." 
In  1861  another  Frenchman,  Dufour,  succeeded  in  raising  globules 
of  water  immersed  in  oil  to  a  temperature  of  175 °C. 


Experiment 

To  determine  the  variation  of  the  boiling  point  with 
the   pressure,    (1)  by  the  static  method;    (2)  by  the 
dynamic  method. 

DIRECTIONS. — 1.  The  apparatus  used  in  the  static  method 
is  shown  in  Fig.  85.  The  bulb  J9,  originally  open  at  c,  was  first 
half  filled  with  mercury.  The  long  arm  (about  5  mm.  in  diam- 
eter), also  originally  open  at  the  top,  was  then  exhausted  and 
inclined  till  the  mercury  completely  filled  it  up  to  a  point  at  which 
it  had  been  drawn  down  to  capillary  dimensions.  The  tube  was 
then  sealed  off  at  this  point,  so  that  whe.n  the  instrument  was 
vertical  the  difference  between  the  levels  of  the  mercury  in  the  bulb 
and  in  the  tube  was  equal  to  the  barometric  height. 
the  bulb  Water  was  then  inserted  at  c  and  boiled  until  the  air  was 
all  driven  from  the  bulb,  when  the  opening  at  c  was 
sealed  off.  Since,  then,  only  water  and  water  vapor  exist  above 
the  mercury  in  the  bulb,  the  difference  between  the  levels  in  the 


158 


MOLECULAR    PHYSICS    AND    HEAT 


The 
readings. 


bulb  and  in  the  tube  gives  at  once  the  pressure  of 
the  saturated  water  vapor  in  the  bulb.  It  is,  there- 
fore, only  necessary  to  vary  the  temperature  of 
the  bulb  in  order  to  obtain  the  curve  expressing 
the  relation  between  the  temperature  and  the  pres- 
sure of  saturated  water  vapor. 

The  whole  bulb  is  to  be   placed  in  a  jar  of 
water  whose  temperature  is  first  to  be  lowered, 
by  the  insertion  of  ice,  to  nearly  0°C., 
and  then  slowly  raised  to  about  50 °C. 
by  pouring  in  hot  water  and  siphoning 
off  the  cold.    At  about  50°  it  is  well  to  replace 
the  glass  jar  by  a  metal  pail  and  thenceforth  to 
heat  slowly  to  100°  by  means  of  a  Bunsen  flame. 
Between  0°  and  about  70°C.  readings  of  the  pres- 
sure are  to  be  taken  at  intervals  of   10°  to  12°, 
between  70°  and  100°  at  intervals  of  about  4°.  The 
water  must  be  very  vigorously  stirred  throughout 
the  experiment',  and  the  temperature  should  be 
held  constant  for  at  least  one  minute  before  a 
reading  is  taken. 

The  thermometer  readings  must  all  be    cor- 
rected. (1)  for  the  errors  of  the  instrument  itself, 

and  (2)    f°r  the   lengtn  of    the  exposed 

thread  of   mercury.     If   the  first  cor- 
thermometer.  rection   is  to  be  made  accurately,  the 
FIGURE  85  thermometer  should  be  one  which  has  been  com- 

pared with  a  standard,  as  in  Ex.  XVI.  If,  how- 
ever, this  comparison  has  not  been  made,  the  corrections 
may  be  obtained  with  a  moderate  degree  of  accuracy  by  observ- 
ing the  corrections  at  the  freezing  and  boiling  points  and 
interpolating  between  these  points  for  the  corrections  at  other 
temperatures.  Thus,  suppose  that  when  wrapped  in  one  layer  of 
flannel  and  packed  in  shaved  ice  over  which  a  little  water  has 
been  poured,  the  thermometer  reads  -0.2°,  and  that  when  com- 
pletely immersed  to  the  top  of  the  thread  in  a  steam  bath  upon  a 
day  on  which  the  boiling  point  of  water  should  be  99.4°,  the  reading 
is  100.5°.  The  corrections  at  0°  and  100°  are  then  +.2  and  -1.1 
respectively.  If  these  two  corrections  be  plotted  as  ordinates 


Corrections 


TENSION"    OF    SATURATED    VAPOR 


159 


upon  a  horizontal  line  representing  temperatures,  and  if  the  ends 
of  these  ordinates  be  connected  by  a  straight  line,  as  in  Fig.  86, 
the  corrections  for  intermediate  temperatures  may  be  read  off 


FIGURE  86 


Correction 
for  the  ex- 
posed thread. 


from  this  curve.     Thus,  in  this  case,  the  correction  at  10°  is-f.l, 
at  40°  it  is  -.3,  at  95°  it  is  -1.05,  etc. 

The  correction  for  the  exposed  thread  is  obtained  by  adding  to 
the  observed  temperature  .OOOWl(t  —  ^0),  in  which  /  is  the  length 
in  degrees  of  the  exposed  thread  of  mercury,  t  the 
observed  temperature  corrected  according  to  (1),  tQ  the 
mean  temperature  of  the  exposed  column  obtained  from 
a  second  thermometer  whose  bulb  hangs  about  the  middle  of  this 
column,  and  .00016  the  apparent  expansion  coefficient  of  mercury 
in  glass  [.000181  (=  coef.  of  Hg)  -  .000025  (=  coef.  of  glass) 
-  .00016,  approximately]. 

In  order  to  compare  the  results  with  tabulated  values  of  vapor 
pressures,  it  is  necessary  to  express  all  pressures  in  terms  of  col- 
umns of  mercury  at  0°C.  This  correction  is  made  as 
usual  by  multiplying  the  observed  heights  by  the  ratio 
of  the  densities  of  mercury  at  the  mean  temperature 
of  the  observed  column  and  at  0°  (see  Appendix). 
This  correction  may  be  made  very  roughly,  for  it  is  only  at  the 
higher  temperatures  that  it  amounts  to  more  than  the  observa- 
tional errors.  The  mean  temperature  of  the  column  may  be 
taken  from  a  third  thermometer  hung  near  its  middle  point. 

The  observed  pressure  will  need  a  still  further  cor- 

Correctfon  .,,  _  .  .      , 

for  capillary   rection  on  account  of  the  capillary  depression  of  the 

depression. 

mercury  m  the  tube.     This  correction  may  be  taken 
from  the  table  in  the  Appendix. 

2.  The  dynamic  method  consists  in  the  direct  observation  of 
the  temperatures  of  the  steam  which  rises  above  a  boiling  liquid 


Reduction 
of  the  ob- 
served pres- 
sures to  o°C. 


160 


MOLECULAR   PHYSICS   AND   HEAT 


made  to  boil  under  varying  pressures.  In  Fig.  87,  A  represents 
an  air-tight  metal  boiler,  which  may  be  replaced  if  need  be  by  a 
simple  long-necked  glass  flask.  B  is  a  condenser  through  which 


\D 


FIGURE  87 


a  slow  current  of  water  is  passed  from  the  tap  TI  .  It  is  only  by 
The  water  virtue  of  the  immediate  condensation  of  the  steam 
as  it  forms  that  the  pressure  within  the  boiler  can 
be  kept  constant.  C  is  an  air-tight  chamber  of  sufficient  capac- 
ity (in  this  case  about  4  liters)  to  prevent  irregularities  in 
the  boiling  from  producing  appreciable  changes  in  the  pressure. 
The  only  other  essential  features  of  the  apparatus  are  a  manom- 
eter D  and  any  sort  of  an  air  pump.  That  shown  in  the  figure 
is  a  Bunsen  water  pump,  such  as  may  be  very  conveniently  used 


TENSION    OF   SATUBATED    VAPOR  161 

in  any  laboratory  which  is  supplied  with  running  water.  As  soon 
as  the  tap  Tz  is  opened,  a  jet  of  water  rushes  through  the  orifice 
at  o  and  draws  with  it  the  air  from  the  chamber^,  thus  exhaust- 
ing any  vessel  with  which  E  is  connected.  H  is  a  trap  to  prevent 
water  from  being  sucked  back  into  the  manometer  when  the 
pump  is  stopped.  F  is  a  three-way  stopcock  (see  Fig.  77b, 
p.  131)  by  means  of  which  the  boiler  may  be  put  into  connection 
either  with  the  pump  or  with  the  air,  or  cut  off  entirely  from  out- 
side communication.  G  is  also  a  three-way  cock,  which  is  inserted 
so  that  it  may  be  unnecessary  to  disconnect  the  boiler  if  it  is 
desired  to  use  the  water  pump  for  exhausting  purposes  in  other 
experiments. 

First,  start  the  circulation  in  the  condenser,  then  turn  F  until 
A  is  in  communication  with  the  air,  and  start  the  water  to  boil- 
*n&'  After  the  conditions  have  become  stationary, 
rea(j  the  barometer  and  the  boiling  temperatures. 
Then  turn  F  so  that  the  boiler  communicates  with  the  pump,  and 
allow  the  water  to  run  until  a  difference  of  5  or  10  cm.  has  been 
produced  in  the  arms  of  the  open  mercury  manometer  D.  In  this 
regulation  of  pressure,  strive  to  duplicate  as  nearly  as  possible  some 
pressure  used  in  the  static  method.  Next  close  ^entirely,  stop 
the  pump,  and  after  waiting  about  two  minutes  take  several 
observations  of  the  new  boiling  point  and  the  corresponding 
pressure.  Then  again  start  the  pump,  put  the  boiler  into  con- 
nection with  it,  and  reduc'e  the  pressure  to  some  second  value  used 
in  1.  Continue  in  this  way  until  the  boiling  temperature  has 
fallen  to  about  75 °C.  In  this  method  the  observations  can  not 
be  conveniently  carried  to  temperatures  lower  than  75°,  because, 
with  much  further  exhaustion,  the  difficulty  of  boiling  with 
bumping  is  encountered.  By  attaching  a  bicycle  pump  to  F,  the 
boiling  point  for  pressures  somewhat  higher  than  76  cm.  can  be 
investigated.  Correct  the  thermometer  readings  exactly  as  in  1, 
and  tabulate  results  in  the  form  shown  in  the  Record.  Finally, 
it  is  required  to  plot  in  the  note-book  a  full-page  curve  in  which 
temperatures  are  represented  by  abscissae,  and  pressures  by  ordi- 
nates.  The  book  values  are  to  be  indicated  by  dots,  the  values 
obtained  in  1  by  crosses,  and  those  obtained  in  2  by  circles. 
The  smooth  curve,  which  comes  as  nearly  as  possible  to  touching 
all  these  points,  is  the  curve  required. 


162  MOLECULAR    PHYSICS    AND    HEAT 

Record 

IST  METHOD  2D  METHOD 

' ,  , • .,  Book 

TEMPERATURES        PRESSURES       TEMPERATURES       PRESSURES      val- 

cor-       cor-  pres- 

rected  rected  red'c'd    cor-  cor-        baro-  mano-  total    sures 

obs'd  for  (1)  for  (2)      obs'd  to  0°    rected       obs'd  rected     meter  meter  red'c'd 


Problems 

1.  Assume  that  the  laws  which  hold  for  ideal  gases  hold  also 
for  vapors  up  to  the  very  point  of  saturation,  i.e.,  assume  that 
equation  (146)  is  applicable  to  saturated  vapors.     Then,  with  the 
aid  of  the  known  density  of  air  at  0°,  76  cm.,  viz.,  .001293,  the 
density  of  water  vapor  in  terms  of  air,  viz.,  .624,  equation  (146), 
and  the  above  values  for  the  pressures  of  saturated  water  vapor, 
calculate  the  densities  of  saturated  water  vapor  at  10° C.,  at  40° 
C.,  at  70°C.,  at  100°C.,  and  compare  with  the  observed  densities 
given  in  the  table  in  the  Appendix.     The  results  will  show  how 
closely  the  gas  laws  apply  even  to  saturated  vapors. 

2.  In  a  uniform  barometer  tube  in  which  the  mercury  stands 
but  40  cm.  high,  the  space  above  the  mercury  is  40  cm.  long,  and 
contains  at  first  only  dry  air.     A  few  drops  of  ether  are  then 
introduced  into  the  tube.     If  the  tension  of  saturated  ether  vapor 
at  the  temperature  of  the  room  is  30  cm.,  find  to  what  height 


TENSION    OF    SATURATED    VAPOR  163 

above  the  mercury  in  the  cistern  the  mercury  in  the  tube  will 
ultimately  fall. 

3.  If  the  bulb  of  the  apparatus  shown  in  Fig.  85  were  gradually 
heated  above  100°,  would  any  temperature  ever  be  reached  at 
which  the  water  within  the  bulb  would  be  observed  to  boil? 

4.  Explain  why,  from  the  standpoint  of  the  kinetic  theory,  a 
lower  temperature  can  be  reached  by  fanning  an  open  vessel  of 
ether  than  by  fanning  an  open  vessel  of  water. 


XIX 

HYGKOMETRY 

Theory 

Hygrometry  is  that  branch  of  Physics  which  relates  to  the 
study  of  the  water  vapor  contained  in  the  earth's  atmosphere. 
The  urvose  From  the  considerations  presented  in  Ex.  XVIII,  it  is 
metrffobser-  evident  that,  were  it  not  for  the  presence  of  the  air,  the 
vations.  earth  would  always  be  covered  with  this  vapor  in  a 
saturated  condition,  and  precipitation  in  the  form  of  fog,  dew,  or 
rain  would  accompany  every  fall  in  temperature,  however  slight. 
But  the  presence  of  air  so  retards  the  process  of  evaporation  that 
even  in  the  immediate  neighborhood  of  lakes  or  oceans  the  condi- 
tion of  saturation  does  not  usually  exist.  Hence  it  is,  that  precip- 
itation often  fails  to  occur  even  when  the  thermometer  falls 
suddenly  through  many  degrees.  Nevertheless,  a  knowledge  of 
the  hygrometric  state,  i.e.,  the  state  of  dryness,  or  wetness,  of 
the  atmosphere,  or,  what  amounts  to  the  same  thing,  a  knowledge 
of  the  number  of  degrees  through  which  the  temperature  must 
fall  before  dew  can  form,  is  of  considerable  importance  not  only 
for  scientific  but  also  for  practical  purposes,  such,  for  example, 
as  the  forecasting  of  the  probability  of  frost,'  or  the  maintenance 
within  green-houses,  drying-rooms,  sick-rooms,  and  dwellings,  of 
^ suitable  climatic  conditions. 

The  four  Tke  four  quantities  involved  in  hygrometric  deter- 

qumSStaF    minations  are: 

sought.  x.  The  density  of  the  water  vapor  in  the  air,  i.e., 

the  weight  in  grams  of  the  water  vapor  contained  in  1  cc.  of 
space.  This  is  usually  called  the  absolute  humidity.  It  is  here 
represented  by  the  letter  d. 

2.  The  relative  humidity  or  the  degree  of  saturation.  This  is 
represented  by  the  letter  r,  and  is  defined  as  the  ratio  between  the 
density  of  the  water  vapor  existing  in  the  atmosphere  at  any  given 
time,  and  the  largest  donsity  which  it  could  possibly  have  at  the 

164 


HYGROMETRY  165 

existing  temperature;  i.e.,  if  D  represent  the  density  of  saturated 
water  vapor  at  the  existing  temperature,  then  the  relative  humid- 
ity r  is  given  by  r  =  —  -  Since  the  Desalts  of  Problem  1,  page  162, 

have  shown  that  at  ordinary  temperatures  the  density  of  saturated 
water  vapor  can  be  calculated  with  sufficient  accuracy  from  the 
pressure  which  it  exerts  (obtained  in  Ex.  XVIII),  D  may  always 
be  considered  a  known  quantity  (see  also  Appendix,  table  6). 

3.  The  dew-point  T,  or  the  point  to  which  the  temperature 
must  fall  in  order  that  the  water  vapor  existing  in  the  atmosphere 
may  be  in  the  saturated  condition.     Of  course,  as  soon  as  the 
temperature  falls  below  this  point,  condensation  must  ensue. 

4.  The  tension  or  pressure,  p,  which  the  wafer  vapor  in  the  air 
exerts  at  the  existing  temperature. 

As  will  presently  appear,  the  experimental  determination  of 
any  one  of  these  four  quantities  taken  in  connection  with  the 
pressure-temperature  curve  of  a  saturated  vapor  (see  Ex.  XVIII), 
suffices  for  the  calculation  of  all  the  rest. 

The  first  attempt  to  construct  an  instrument  for  measuring 
hygrometric  conditions  was  made  about  1600,  when  an  Italian 
named  Sanctorius  devised  what  is 

Absorption  . 

hygrome-  now  known  as  an  absorption  hygrom- 
eter, an  instrument  usually  asso- 
ciated with  the  name  of  de  Saussure,  a  Genevan, 
who  brought  it  into  prominence  in  1783.  It 
is  well  known  that  many  organic  substances 
expand  with  an  increase  in  the  dampness  of 
the  atmosphere.  De  Saussure's  hygrometer 
consisted  of  a  human  hair  attached  as  in  Fig. 
88,  so  that  changes  in  its  length  caused  a 
pointer  to  move  over  a  scale  which  was  con- 
structed by  marking  the  position  of  the  pointer 
in  a  saturated  atmosphere  100,  its  position  in 
a  perfectly  dry  atmosphere  0,  and  then  divid- 
ing the  intervening  space  into  100  equal  parts. 
The  position  of  the  pointer  at  any  time  was  FIGURE  88 

assumed   to   indicate   directly   the    degree   of 
saturation  of  the  atmosphere  (r).     These  instruments  are  still  in 
common  use,  but  they  are  now  always  graduated  empirically  by 


166  MOLECULAR    PHYSICS   AND    HEAT 

comparison  with  a  dew-point  hygrometer  (see  below);  for  careful 
experiments  made  by  Regnault  in  1845  showed  that  instruments 
constructed  as  above  agree  accurately  neither  with  one  another 
nor  with  the  indications  of  dew-point  hygrometers.  Further,  it  is 
found  that  a  given  absorption  hygrometer  does  not  remain  com- 
parable even  with  itself  for  any  long  interval  of  time.  Hence  its 
indications  can  only  be  relied  upon  if  it  is  frequently  recalibrated. 
Accurate  measurements  in  hygrometry  began  with  the  intro- 
duction by  the  Englishman  Daniell,  in  1820,  of  the  dew-point 

hygrometer,  the  essential  principle  of  which  had  been 
hewrometer  empl°ye(l  by  the  Frenchman  Le  Eoy  as  early  as  1751. 

This  instrument  in  one  or  another  of  its  numerous 
modifications  has  become  the  standard  of  comparison  for  the 
testing  and  graduation  of  all  other  hygrometers.  It  consists 
essentially  of  a  polished  metal  tube,  the  temperature  of  which  is  in 
some  way  lowered  until  dew  is  observed  to  form  upon  its  surface. 
From  this  temperature  of  condensation  T,  it  is  possible  to  deter- 
mine all  the  other  hygrometric  constants. 

Thus  the  pressure^  is  obtained  at  once  from  T  and  the  pres- 
sure-temperature curve  of  a  saturated  vapor.  It  is  simply  the 
TO  obtain  pressure  of  saturated  water  vapor  at  the  temperature  T. 

For,  although  the  cooling  of  the  layers  of  atmosphere 


™romrtheP°r  wo^cn  are  ^n  contact  with  the  metal  surface  causes  an 
dew-point.  increase  in  the  density  both  of  the  air  and  of  the  water 
vapor  of  which  these  layers  are  composed,  yet,  since  the  baro- 
metric pressure  is  in  no  way  affected  by  the  cooling,  it  is  evident 
that  the  pressure  both  of  the  air  and  of  the  water  vapor  within 
these  layers  must  remain  precisely  the  same  as  outside,  where  no 
cooling  takes  place.  The  beginning  of  precipitation  means  only 
that  in  the  layers  adjoining  the  surface  the  density  and  pressure 
corresponding  to  saturation  have  been  reached.  If,  then,  the 
pressure  within  these  layers  is  the  same  as  outside,  it  is  clear  that 
table  6  gives  the  correct  value  of  p. 

Not  so,  however,  with  d.  The  table  gives,  indeed,  the  value 
of  this  quantity  within  the  cooled  layer,  but  the  density  at 
a  distance  is  the  density  within  the  cooled  layer  multiplied 

,    Trr  273°  +  °n 

by  —  =  —  —  ^  —  —  ;  for  when  pressure  remains  constant,  density  is 
inversely  proportional  to  absolute  temperature.  Instead  of  obtain- 


HYGROMETRY 


167 


ing  d  in  this  way  from  the  table,  it  may  be  directly  calculated 
from  p  by  the  ordinary  gas  laws,  viz.,  by  (146).  Thus,  since 
the  densitj  of  air  at  0°,  760  mm.,  is  .001293,  and 
gince  under  like  condition  of  temperature  and  pres- 
e  density  °f  water  vapor  in  terms  of  air  is  .624, 
point.  (146)  becomes,  when  dai,  2h  and  TI  are  replaced  by 

d,  p  and  Ttt  and  da.2,  2h  and  T2  by  .001293  x  .624,  760  and  273, 


ro  obtain 
ofewatertv 
lthedewrom  stire 


.001293  x. 624x273     _  .  00029  p 
760  .  Tt        ~P~    ~^T~1 


(149) 


in  which  p  must,  of  course,  be  expressed  in  mm.  of  mercury, 
since  the  barometric  pressure  has  been  so  expressed.  This  exten- 
sion to  unsaturated  vapor  of  the  laws  which  hold  rigorously  only 
for  ideal  gases  must  be  permissible  in  practice,  since  the  results  of 
Problem  1,  page  162,  have  shown  that  at  ordinary  temperatures 
equation  (149)  may  be  applied  without  appreciable  error  even 
to  saturated  water  vapor.  The  rela- 
tive humidity  r  can  be  at  once  obtained 
from  either  p  or  d  and  table  6;  for 
d  p 


r  = 


One  of  the  most  perfect  forms  of 
the  dew-point  hygrometer  is  due  to  the 
Frenchman  All uard  (1880), 
is  shown   ir 


SSmeter.      The    nickel    tube    A)    uPon 

which  the  dew  is  formed,  is 
about  2  cm.  in  diameter,  and  has  one 
flat,  highly  polished  side  which  is  placed 
in  close  juxtaposition  to  a  strip  B  of 
equally  well  polished  nickel  upon  which 
no  dew  is  formed.  Tube  A  is  filled 
with  ether,  the  temperature  of  which 
may  be  lowered  by  causing  a  current  of 
air  to  bubble  through  it.  This  is 
accomplished  by  means  of  an  aspirator 
attached  by  a  rubber  tube  to  C.  A 
bellows  attached  to  F  serves  the  pur- 
pose equally  well.  The  experimenter 


FIGURE  89 


168 


MOLECULAR    PHYSICS    AND    HEAT 


The  chem- 
ical hygrom- 
eter. 


sits  at  a  distance  of  10  or  12  feet,  so  that  the  moisture  from  his 
breath  or  body  may  not  affect  the  result,  and  observes  the  tube  and 
thermometer  through  a  telescope.  At  the  instant  at  which  A  begins 
to  look  cloudier  than  B,  he  takes  the  temperature  indicated  by  the 
thermometer  E.  He  then  stops  the  current  of  air  and  observes 
again  the  temperature  at  which  the  cloudiness  disappears  from  A. 
With  a  little  practice  the  temperatures  of  appearance  and  disap- 
pearance of  the  dew  can  be  made  to  approach  to  within  .1°G. 
The  mean  of  these  two  temperatures  is  taken  as  the  dew-point. 
This  form  of  instrument  should  not  be  used  in  rapidly  moving 
air,  for  then  the  layers  of  air  which  are  in  contact  with  A  are 
removed  before  they  can  take  up  the  temperature  of  the  nickel, 
and  in  consequence  the  observed  dew-points  are  too  low. 

The  indications  of  a  dew-point  hygrometer  may  be  very  nicely 
checked  by  means  of  the  chemical  method,  first  used  for  hygro- 
metric  determinations  by  the  Swiss  chemist  Brunner  in 
1844.  It  consists  in  slowly  drawing  a  known  volume 
of  air  V  through  drying-tubes,  preferably  of  anhydrous 
phosphorus  pentoxide,  and  measuring  the  increase  w  in  the  weight 
of  the  tubes.  Let  v  represent  the  volume  of  water  which  has 
been  drawn  out  of  the  aspirator  R  during  the  experiment  (see 
Fig.  90).  The  gas  which  has  replaced  this  water  consists  of  the 
perfectly  dry  air  which 
has  emerged  from  the 
drying-tubes  and  the 
water  vapor  which  has 
formed  from  the  water 
in  R.  Since  this  vapor 
may  be  assumed  to  be 
saturated,  the  pressure 
which  it  exerts  is  P, 
the  pressure  of  satu- 
rated vapor  corresponding  to  the  temperature  of  the  room.  Hence 
the  pressure  exerted  by  the  air  alone  which  is  within  R  is  H—  P, 
H  being  the  barometric  pressure.  When  this  same  air  was 
outside,  it  exerted  a  pressure  H—p,  p  being,  as  above,  the 
pressure  exerted  by  the  water  vapor  in  the  outside  air.  Hence 
the  volume  which  the  air  in  R  occupied  before  it  entered 
the  drying-tubes,  i.e.,  the  volume  of  air  Fin  which  the  weight 


FIGURE  90 


HY  GEOMETRY 


169 


w  of  water  vapor  was  contained,  is  given  by  (see  Boyle's  Law) 

V  H-P 
v  ~  H-  p 
w 


(150) 


From  (149),  (150),  and  the  relation  —  =  d,  there  results 


.00029;? 


H-P 
H-p 


(151) 


an  equation  which  contains  only  one  unknown  quantity,  viz.,  7?. 
After  p  has  been  determined  from  (151),  d  and  r  may  be  found  as 
above,  while  r  is  taken  from  table  6  ;  i.e.,  it  is  the  temperature-  of 
saturation  of  water  vapor  corresponding  to  the  pressure  p.  In 
case  P  and  p  have  nearly  the  same  value,  (150)  gives  V-  v,  in 


which  case  d  is  obtained  at  once  from  d  =  -,  and  p  is  then  found 

v 

from  (149). 

This  chemical  method  leaves  nothing  to  desire  in  the  matter  of 
accuracy,  but  since  an  observation  usually  requires  from  1  to  3 
hours,  the  result  is,  of  course,  only  a  mean  value  of 
the  humidity  during  this  time.     It  is  little  employed 
in  practice. 

The  instrument  which  is  now  most  extensively 
used  in  meteorological   observations  was  first  con- 
ceived by  the  Scotch  physicist  John  Leslie 


The  wet-and-    .  T ,  .  . . 

dry  huib  m  1790.  It  was  given  its  present  form 
by  August,  in  Berlin,  about  1825.  It  is 
called  the  wet-and-dry  bulb  hygrometer,  and  con- 
sists of  two  sensitive  thermometers  mounted  side  by 
side,  one  of  which  has  its  bulb  wrapped  in  muslin, 
which  is  kept  moist  by  means  of  a  cotton  wick 
immersed  in  a  water  reservoir  c  (see  Fig.  91).  The 
temperature  t'  indicated  by  the  wet-bulb  thermom- 
eter is  always  lower  than  the  temperature  t  shown 
by  its  dry-bulb  neighbor,  unless  the  water  vapor  in 
the  air  is  already  in  a  state  of  saturation,  for  the 
evaporation  which  otherwise  takes  place  produces 
cooling  (see  page  155).  The  amount  of  this  cool- 
ing evidently  increases  with  the  rapidity  of  evaporation,  which  is 
in  turn  directly  proportional  to  the  dryness  of  the  atmosphere, 


FIGURE  91 


170  MOLECULAR    PHYSICS   AND    HEAT 

i.e.,  to  D  —  d,  or,  what  amounts  to  the  same  thing,  to  P  —  p.  But 
it  also  depends  upon  the  density  of  the  air  into  which  the  evapo- 
ration takes  place,  i.e.,  upon  the  barometric  height  H,  as  well  as 
upon  the  velocity  of  the  air  currents  (see  page  155).  A  long 
series  of  comparisons  of  this  instrument  with  the  dew-point  hygrom- 
eter has  given  for  air  in  moderate  motion  the  empirical  formulae 

(for  f  above  0°)  (for  t'  below  0°) 

p  =  P-0. 00080  H(t-t'),  and  p  =  P-0. 00009  H  (t-tf),  (152) 
in  which  p,  P  and  H  are  expressed  in  mm.  of  mercury  and  (t  —  t') 
in  degrees  centigrade.  When^?  has  been  determined  from  this 
equation,  d  is  given  by  (149),  and  then  r  and  T  from  table  6. 
For  air  at  rest,  the  numerical  quantities  in  (152)  are  considerably 
too  small.  In  order  to  make  (152)  applicable  to  indoor  observa- 
tions, it  is  recommended  to  hang  the  instrument  from  a  long  cord 
and  to  let  it  swing  as  a  pendulum  for  several  minutes  before  tak- 
ing the  readings.  Hygrometer  makers  usually  specify  carefully 
the  conditions  under  which  instruments  of  this  type  are  to  be 
used,  and  furnish  with  them  empirical  tables. 


Experiment 

To  compare  the  indications  of  a  dew-point  and  a 
wet-and-dry  bulb  hygrometer. 

Place  the  Alluard  hygrometer  in  a  room  which  is  free  from 
evaporating  water,  pour  ether  into  G  (Fig.  89)  until  the  liquid 
surface  is  above  the  window  in  A,  turn  the  polished 
metal  face  into  as  favorable  a  light  as  possible,  set  the 
observing-stand,  telescope,  and  aspirator  at  a  convenient  distance, 
and  take,  first,  a  rough  observation  of  the  dew-point.  In  subse- 
quent observations  regulate  the  evaporation  of  the  ether  so  that 
the  temperature  falls  very  slowly  in  the  neighborhood  of  the  point 
sought,  and  take  the  reading  of  the  thermometer  E  when  the  first 
cloudiness  begins  to  show  upon  A.  In  taking  observations  with 
a  rising  temperature,  an  occasional  bubble  may  be  allowed  to  pass 
through  the  ether  in  order  to  keep  it  stirred.  Tabulate  results 
as  in  the  Record.  Then  bring  into  the  room  a  wet-and-dry  bulb 
hygrometer,  and  take  a  set  of  observations  in  the  manner  indicated 
in  the  Theory. 


No.  of 
obs'n 


HYGROMETRY 

Record 


DEW-POINT  HYGROMETER, 


Room 
tern.  (E') 


Dew 
app'd  (E) 


Dew 
disapp'd  (E) 


Mean 


P  = 


.-.  d  = 


171 


Wet-and- 
dry  bulb 

H=  — 


Problems 

1.  When  the  relative  humidity  is  .47  at  21  °C.,  what  will  be 
the  dew-point? 

2.  If  the  temperature  of  the  air  at  sundown  on  a  clear  day  be 
10°,  and  if  the  wet-bulb  thermometer  read  8°C.,  at  what  tem- 
perature will  dew  form?     Need  there  be  fear  of  frost  during  the 
night?     (Bar.  ht.  750  mm.) 

3.  If  the  wet-bulb  thermometer  of  Problem  2  had  read  4.5°C., 
what  would  have  been  the  dew-point?     In  this  case  frost  would 
have  been  almost  certain.     Why? 

4.  Dry  air  at  18°,  755  mm.,  weighs  .001205  gm.  per  cc.     Find 
the  density  of  the  atmosphere  at  this  temperature  and  pressure 
when  the  dew-point  is  10° C. 


XX 

ARCHIMEDES'   PRINCIPLE 

Theory 

The  law  which  asserts  that  the  loss  in  loeight  experienced  by 

any  body  ivhen  immersed  in  a  fluid  is  equal  to  the  iveight  of  the 

displaced  fluid,  was  discovered  by  the  immortal  Greek 

Proof  of  law          *  J          '     .  J.  . 

ofArcM-  philosopher  Archimedes,  who  perished  in  the  siege  of 
Syracuse  in  212  B.C.  Unlike  many  of  the  laws  which 
have  preceded,  it  is  not  an  approximation,  nor  is  it  primarily 
empirical,  experiment  having  only  served  to  confirm  results  which 
follow  with  certainty  from  theory.  The  work  of  Archimedes  was 
not  known  in  the  middle  ages,  and  the  law  was  rediscovered  in 
1586  by  the  Flemish  scientist  Stevin,  who  advanced  for  it  the 
following  proof :  Within  a  body  of  fluid,  isolate  in  thought  some 
mass  by  means  of  any  imaginary  bounding  sur- 
face 8  (see  Fig.  92)4.  Since  the  mass  of  liquid 
within  this  boundary  is  in  equilibrium,  its 
weight  must  be  neutralized  by  forces  whose 
existence  is  due  to  the  surrounding  liquid. 
But  these  forces  which  are  exerted  by  the 


surrounding  liquid  upon  the  surface  S  depend 
only  upon  the  conditions  which  exist  outside  of  S,  and  are  wholly 
independent  of  the  nature  of  the  substance  within  S.  Hence 
any  immersed  body  whatever  which  has  the  surface  S  must  be 
buoyed  up  by  forces  the  resultant  of  which  is  equal  to  the  weight 
of  the  displaced  liquid. 

It  follows  from  this  law  that  when  a  body  is  balanced  upon 
scales  in  air,  the  balancing  weights  do  not  accurately  represent 
Application  ^°e  ^nie  weign^  °^  ^ne  body,  i.e.,  its  weight  in  vacuo. 
of. ArS  -^or  k°th  ^e  body  and  the  weights  are  buoyed  up  by 
™etermina-e  ^e  a^r'  an(^  since  in  general  the  volume  of  air  dis- 
^rlct  placed  by  the  body  is  not  the  same  as  that  displaced 
weight.  by  the  weights,  the  buoyant  effects  upon  the  two  sides 
must  be  different.  However,  with  the  aid  of  the  "principle  of 
moments"  the  true  weight  X  can  be  easily  obtained  from  the 
apparent  weight  JF(i.e.,  the  weight  of  the  weights),  the  volume 

172 


r 


ARCHIMEDES7   PRINCIPLE  173 

Fof  the  body,  the  volume  V  of  the  weights  and  the  density  <r  of 
air.  For,  since  the  resultant  downward  force  on  the  side  of  the 
body  is  (X-  F<r)  grams,  and  that  on  the  other  side  (W-  V'a) 
grams,  the  equation  of  balance  for  equal  balance  arms  becomes 

X-  V<r=  W-  V'a-    or     X=  PF+(F-  F>.      (153) 

Hence  the  so-called  air  correction,  i.e.,  the  correction  which  must 
be  applied  to  the  apparent  weight  W  in  order  to  obtain  the  true 
weight  X,  is  simply  the  difference  between  the  weight  of  air  dis- 
placed by  the  body  and  that  -displaced  by  the  weights.  This 
correction  is  evidently  positive  if  V  >  V  ,  negative  if  F  <  V. 
Since  the  density  of  the  weights  is  always  known  (for  brass  it  is 

W 

8.4),  it  is  usually  convenient  to  replace  F'  by  —  :,   and  to  write 

O.4: 

(153)  in  the  form 

(154) 


Or  again,  if  the  density  d  of  the  body  happens  to  be  roughly 
known,  (153)  takes'  the  approximately  correct  form  (since,  on 
account  of  the  smallness  of  the  correction  term,  Xis  usually  very 
nearly  equal  to  JF), 


In  deducing  the  above  expressions  for  a  correct  weight  X,  the 
balance  arms  were  assumed  to  be  equal.  Although  this  is,  of 
course,  seldom  rigorously  the  case,  it  was  shown  on  p.  39  that 
the  error  arising  from  any  inequality  can  be  completely  eliminated 
by  taking  a  mean  of  the  weighings  made  on  both  pans.  Hence,  in 
order  to  obtain  a  very  accurate  weight,  it  is  necessary  first  to  find 
W  by  means  of  a  double  weighing,  and  then  to  apply  to  W  the  air 
correction  as  shown  in  (153). 

In  case  the  ^quantity  sought  is  a  small  increase  or  decrease  in 
weight,  as  in  Exs.  XV  and  XVII,  the  rigorous  process  described 
in  this  section  is  not  to  be  recommended;  for  neither  the  error  in 
the  balance  arms  nor  the  air  correction  due  to  the  weights  is  in 
such  cases  likely  to  be  an  appreciable  quantity.  For  most  pur- 
poses single  uncorrected  weighings,  in  which,  however,  the  body 
weighed  always  hangs  from  the  same  balance  arm  (see  p.  118),  are 
sufficiently  accurate. 


174  MOLECULAR    PHYSICS    AND    HEAT 

Archimedes'  principle  also  furnishes  the  most  convenient  and 
accurate  method  of  determining  densities.  For,  if  any  body, 
regular  or  irregular,  whose  absolute  weight  is  X  grams, 
ke  f°un(i  to  lose  L  grams  when  weighed  in  water  of 
densitv  P»  ^  is  evident  at  once  from  the  statement  of 
Archimedes'  principle  that  the  volume  of  the  body  is 

denxity  of          £ 

a  solid.  — ?  an(j  hence  that  its  density  d,  which  is  bv  definition 

P 
mass      ,      . 

— = ,  is  given  by 

volume' 

*-£-•£•  (»6) 

P 

But,  in  order  to  find  L  accurately,  an  air  correction  of  the  ordi- 
nary form  must  be  applied  to  the  apparent  loss  of  weight.  For 
let  the  body  whose  true  weight  is  X  and  whose  apparent  weight 
is  W  be  immersed  in  water  and  weighed,  first  when  hung  from  one 
balance  arm,  then  from  the  other,  and  let  this  mean  apparent  weight 
in  water  be  W^.  The  equation  of  balance  for  this  case,  in  which 
the  body  hangs  in  water  while  the  weights  hang  in  air,  is  evidently 

X-  VP  =  Wi-^r.  (157) 

Substituting  the  value  of  X  found  in  (154),  there  results  at  once 

(W— 
V-  Ztj 

which  is  an  equation  of  the  same  form  as  (154),  the  apparent 
weight  W  having  been  simply  replaced  by  the  apparent  loss  of 
weight  (W-WJ. 

The  very  statement  of  Archimedes'  principle  also  suggests  its 

use  for  determining  the  densities  of  liquids.     For,  if  L  represent 

the  loss  of  weight  of  a  body  in  water  of  density  /o,  and 

afPtheCiawn    ^  ^he  loss  of  weight  of    the  same  body  in  another 

medestothe    ^u^  °^  unknown  density  d,  then,  since  the  volume  of 

determina-  J^ 

denrf?  *«/       ^e  b°cly  is  — ,  the  density  of  the  unknown  liquid  is 

a  liquid.  P 

given  by 


ARCHIMEDES'  PRINCIPLE 


175 


The  weighings  in  the  two  liquids  may,  of  course,  be  made  with 
an  ordinary  balance,  precisely  as  above,  but  for  the  sake  of 
rapidity  a  modified  form  of  balance  due  to  Mohr  (see  Fig.  93)  is 
commonly  used. 

The  principal  features  of  this  balance  are  (1)  the  division  of 
one  arm  into  ten  equal  parts,  and  (2)  the  use  of  weights  of  con- 
venient shape  to  hang 
from  any  of  the  ten 
notches,  the  tenth  of 
which  is  the  hook  c. 
Since  each  weight  can 
be  given  ten  differ- 
ent values,  a  very 
small  number  of 
weights  is  required. 
There  are  never  more 
than  five.  Further, 
as  will  presently  ap- 
pear, the  absolute 
value  of  these  weights 
is  of  no  importance 

if  only  their  ratios  are  FIGURE  93 

accurately  represented 

by  the  numbers  1,  .1,  .01,  and  .001  (the  fifth  weight  is  a 
duplicate  of  1,  used  for  the  sake  of  extending  the  range  of 
the  instrument).  For,  suppose  that  the  body  B  is  of  such 
weight  that,  when  hung  in  air  from  the.  hook  c,  it  is  possible 
by  means  of  a  little  adjustment  (see  Experiment)  to  balance  the 
beam  so  that  the  two  points  at  a  are  exactly  together.  Next 
suppose  that  when  B  is  immersed  in  water  as  in  the  figure,  it 
is  necessary,  in  order  to  bring  the  points  together  again,  i.e., 
in  order  just  to  counterbalance  the  buoyancy  of  the  water,  to 
hang  weight  1  and  weight  .1  from  the  hook  c,  and  weights  .01 
and  .001  from  notch  4.  The  loss  of  weight  L  of  the  bulb, 
expressed,  not  in  grams,  but  in  terms  of  weight  1,  is  then  evi- 
dently 1.1044.  Suppose  now  that  when  the  water  is  replaced  by 
another  liquid  of  the  same  temperature,  it  is  found  necessary,  in 
order  to  bring  the  points  again  together,  to  hang  1  from  notch  9, 
.1  from  notch  2,  .01  from  notch  4,  and  .001  from  notch  9.  The 


176  MOLECULAR   PHYSICS   AND    HEAT 

loss  of  weight  of  the  bulb  in  this  liquid,  expressed  as  above  in 
terms  of  1,  is  then  .9249.  Hence  (159)  gives 

j_  .9249 
~ri044p' 

If  p  is  known  d  is  at  once  obtained,  no  matter  what  happens  to  be 
the  weight  of  1.  In  practice  this  weight  is  usually  arranged  to  be 
as  nearly  as  possible  equal  to  the  weight  of  the  water  displaced  by 
the  bulb  at  15°C.  In  this  case  the  reading  for  water  at  this 
temperature  is  evidently  1.0000,  and  if  no  high  degree  of  accuracy 
is  required,  the  reading  of  the  instrument  when  the  bulb  is 
immersed  in  the  unknown  liquid  gives  at  once  the  density  of  that 
liquid.  Eigorously  the  apparent  losses  in  weight  L  and  £', 
obtained  by  means  of  a  Mohr's  balance,  are  subject  to  air  correc- 
tions, but  since  in  practice  the  balance  is  only  used  to  compare 
densities  which  differ  comparatively  little  from  one  another,  the 
influence  of  these  corrections  upon  the  result  is  negligible.  They 
could  be  made,  if  necessary,  by  reducing  L  and  L'  to  grams  and 
then  applying  (158),  F  being  in  this  case  the  volume  of  the  bulb. 


Experiment 

1.  To  compare  a  density  determination  made  upon  an  accurately 
turned  cylinder  of  aluminium  by  means  of  weight  and  dimension 
measurements,   with  a  determination  made  upon  the 
same  cylinder  by  means  of  observations  of  weight  and 
loss  of  weight  in  water.     2.  To  compare  the  results  of  measure- 
ments upon  the  density  of  a  liquid  made  by  a  Mohr's  balance, 
with  those  made  by  an  ordinary  constant-weight  hydrometer. 

DIRECTIONS. — 1.   First  find  the  volume  of  the  cylinder  from 
very  careful  measurements  of  its  dimensions,  made  by 

Volume. 

means  of  calipers. 

Then  suspend  the  cylinder  from  a  very  fine  platinum  wire  and 
make  an  absolute  weighing  in  air  of  the  cylinder  and  wire  as  follows : 
Absolute  CO  Choose  any  convenient  zero  to  which  to  refer  your 
weighing.  weighings,  e.g.,  the  10  mark.* 

*The  true  zero  of  the  unloaded  balance  may  be  determined  if  desired, 
but  this  is  never  necessary  with  a  double  weighing;  for  evidently,  if  the 
arbitrarily  chosen  zero  differs  from  the  true  zero,  the  weighing  upon  one 


ARCHIMEDES'  PRINCIPLE  177 

(2)  Hang  the  body  from  the  left  arm  and,  using  all  the  precau- 
tions mentioned  on  page  116,  find  the  resting  point  JKj,  which 
corresponds  to  such  a  weight  in  the  right  pan  as  will  keep  the 
pointer  swinging  within  three  or  four  divisions  of  the  chosen  zero. 

(3)  Add  a  2  mg.  weight  to  the  lighter  side  and  take  the  new 
resting  point  J?2;    then  at  once  raise  the  arrest  and  compute  the 
sensitiveness. 

(4)  Exchange  the  positions  of  the  cylinder  and  the  weights,  at 
the  same  time  making  and  recording  a  careful  count  of  the  latter, 
exclusive  of  the  two  added  milligrams. 

(5)  Find  the  new  resting  point  7?3;  then  raise  the  arrest  and 
count  again  the  weights  as  they  are  returned  to  their  respective 
compartments  in  the  box  of  weights. 

(6)  Calculate  from  the  sensitiveness  the  corrections  necessary  to 
apply  to  the  counted  weight  upon  each  side,  in  order  to  make 
both  of  the  resting  points  coincide  with  tke- chosen  zero.     Thus, 
if  the  zero  is  10,  the  sensitiveness  2.1,   and   the  resting  point 
when  the  cylinder  is  on  the  right  pan  10.9,  the  correction,  in  this 

case  to  be  subtracted  from  the  counted  weight,  is  -    — —  =  .4. 

/£.  1 

Represent  the  mean  corrected  weight  by  W  and  the  result  obtained 
by  applying  to  W  the  air  correction  (154)  by  X. 

Next  immerse  the  aluminium  cylinder,  in  the  manner  shown 
in  Fig.  94,  in  a  beaker  of  distilled  water  which  has  been  recently 

freed  from  air  by  boiling,  but  which 
in  water.0  has  again  regained  the  temperature  of 

the  room.  Carefully  remove  from  the 
immersed  body  all  bubbles,  even  the  smallest,  by 
means  of  a  camel's  hair  brush,  then  weigh  on  both 
sides,  following  exactly  the  directions  1  to  6  above. 
Represent  the  mean  weight  in  water  by  Tfl5  and 
let  the  result  obtained  by  applying  the  air  cor- 
rection (156)  to  the  apparent  loss  in  weight, 
W-  WH  be  represented  by  L.  FIGUUE 

side  will  be  just  as  much  too  large  as  that  upon  the  other  is  too  small, 
and  the  mean  will  therefore  be  the  same.  If,  however,  a  weighing  is 
made  upon  balances  which  are  known  to  have  arms  so  nearly  alike  that  a 
double  weighing  is  unnecessary,  this  single  weighing  must  be  referred  to 
the  true  zero  obtained  by  the  method  of  oscillations  (see  p.  116). 


178  MOLECULAR    PHYSICS    A^D    HEAT 

In  order  to  obtain  Jf,  the  weight  of  the  cylinder  alone,  the 
weight  w  of  the  suspending  wire  must,  of  course,  be  obtained  and 

subtracted  from  th.e  joint  weight  X  obtained  above. 
the  suspend-  Since  this  is  a  very  small  quantity,  a  single  weighing 

made  upon  one  pan,  and  uncorrected  for  displaced  air, 
is  sufficient.  To  make  this  weighing,  find: 

(a)  The  true  zero  of  the  balance  (see  note,  p.  170); 

(b)  The  resting  point  /*,  when  the  wire  alone  is  on  one  pan, 
and  some  nearly  equal  weight  on  the  other ; 

(c)  The    resting    point   ra   (after  2  mg.   have   been  added  to 
determine  the  sensitiveness  for  this  load) ; 

(d)  The  weight  w  of  the  wire  (obtained  from  the  weight  in  the 
pan  and  the  sensitiveness). 

The  loss  of  weight  L  represents,  of  course,  the  weight  of  water 
displaced  both  by  the  immersed  cylinder  and  the  immersed  por- 
tion of  the  suspending  wire.      Estimate  roughly  the 

Correcting  £  ^  •      -  i       •       i  •         •        -,. 

Lforim-        volume  v  oi  this  immersed  wire  by  measuring  its  diam- 

mersed  wire.  *,•>•,,-,  •  « .        x 

eter  8  (with  the  micrometer  cahper)  and  its  approxi- 
mate length  L  This  volume  v  is  approximately  the  weight  of 
the  displaced  water.  Hence  the  loss  of  weight  L  of  the  cylinder 
is  given  by  L  =  L  —  v. 

2.  Loosen  set  screw  s'  of  the  Mohr's  balance  (Fig. 
93),  and  turn  the  base  until  the  leveling  screw  s  lies  in 
the  vertical  plane  which  includes  the  beam.  Adjust  ver- 
tically at  sf  until  a  convenient  height  for  immersion  is 
reached.  Then  from  the  hook  c  hang  the  float  B,  in  air, 
and  level  by  means  of  s  until  the  two  points  at  a  are 
very  accurately  together.  Then,  by  means  of  the  weights 
bring  the  points  again  together,  first  when  the  float  is 
immersed  in  distilled  water,  then  when  immersed  in  the 
liquid  of  unknown  density,  and  take  the  corresponding 
readings  (see  Theory).  The  density  of  the  water  at  the 
observed  temperature  is  taken  from  the  table  of  water 
densities.  The  temperatures  of  the  two  liquids  compared 
must  be  the  same,  otherwise  a  correction  must  be  applied 
because  of  the  expansion  of  the  float. 

Compare  the  density  given  by  the  Mohr's  balance  with 
the  indication  of  a  direct-reading,  constant-weight  hydrom- 
FIG.  95     eter  (see   Fig.    95).      The  theory  of   this  instrument  is 


ARCHIMEDES'  PRINCIPLE  179 

too  simple  to  require  explanation.  The  reading  is  made  through 
the  liquid,  the  eye  being  placed  as  little  as  possible  beneath  the 
level  of  the  surface.  If  the  instrument  does  not  read  the  correct 
density  of  distilled  water  at  the  observed  temperature,  a  correc- 
tion amounting  to  the  difference  must  be  applied  to  its  indication 
of  the  density  of  the  other  liquid.  For  very  accurate  density 
determinations  with  hydrometers  of  this  sort  it  is  important  that 
the  stem  be  wet  above  the  point  of  contact  with  the  liquid,  since 
otherwise  the  capillary  forces  between  the  liquid  and  the  stem  may 
give  rise  to  very  considerable  errors.  Hence,  before  taking  a  read- 
ing, push  the  instrument  down  below  its  natural  position  of 
equilibrium  and  then  let  it  rise.  If  in  this  operation  the  liquid 
be  observed  to  be  depressed  about  the  stem,  instead  of  elevated, 
the  stem  should  be  carefully  cleaned  with  an  alkali,  e.g.,  potas-t 
sium  hydrate. 


Record 

1.  'Bar.  ht.  =  —  Temp,  of  room  =  —        •    ..%  Density  of  air  = 
Diam.  of  cylinder  1st  obs.  —         2d  —         3d  —  4th  —         mean  = 
Height  of  cylinder        "                                          -  mean  =  --  .  •.  V= 

Weighing  of    cyl.  -f  wire    cyl.  +  wire  wire 

in  air          in  water  alone 

Chosen  zero  =  —  Zero  •    t  *      = 

R!  (cyl.  left)  =  -  Rest.  pt.  r1     =  -    -    P        = 

E2  (cyl.  left)  =  -  Rest.  pt.  r2     =  -     -I 

jR3  (cyl.  right)  =  —  •    .  *.  Sensitiv's        =  —     -5        = 

Counted  wts.  =  —  Counted  wt.    —  —     •    v        = 

.  :  Sensitiv's  (8)  =  -  Cor'd  wt.  (w)  =  - 

Cor'dwt.  left  =  -  -     .:X(=X—w)  =  - 

Cor'd  wt.  right  =  -  ~L(=  L  —  v)    =  - 

Mean(W)  =  -  W,  =  -  d  =  ^  =  -  d  =  ^  = 

True  wt.  (X)  =  —  L  =  —  %  difference  in  d's  = 

2.  Reading  of  Mohr's  balance  in  water  =  —          tern.  =  —  p  = 
Read'g  of  Mohr's  balance  in  unknown  liq.=  —          tern.  =  --  .  •.  d  = 
Hydrom'r  in  water  =  —  .  *.  corr'n  =  —  in  liquid  =  --  .'.  d  (cor'd)  = 


*  t  —  tern,  and  p  =  den.  of  water.  I,  5,  and  v  are  length,  diameter,  and 
volume  of  immersed  portion  of  wire. 


180  MOLECULAR    PHYSICS    AND    HEAT 

Problems 

1.  If  the  length  measurements  made  upon  the  cylinder  were 
3.021,  3.023,  and  3.024  cm.,  and  if  the  diameter  measurements 
were  2.567,  2.563,  2.564,  and  2.562  cm.,  find  what  is  the  first 
uncertain    figure  in  the   number  which  represents  the  volume. 
(Results  should  never  be  recorded  farther  than  to  one  place  beyond 
the  first  uncertain  figure.) 

2.  If  the  weighings  can  all  be  made  with  such  accuracy  that 
the  tenths  ing.  place  is  the  first  place  of  uncertainty,  find  to  how 
many  more  places  of  certainty  the  density  is  given  by  the  loss  of 
weight  method  than  by  the  direct  measurement  method  (weight 
of  cylinder  about  12  gm.). 

3.  Archimedes  discovered  his  principle  when  seeking  to  detect 
a  suspected  fraud  in  the  construction  of  a  crown  made  for  the 
tyrant  of  Syracuse.     It  was  thought  to  have  been  made  from  an 
alloy  of  gold  and  silver  instead  of  from  pure  gold.     If  the  crown 
weighed  1000  gm.  in  air  and  940  gm.  in  water,  find  how  many 
gm.  of  gold  and  how  many  of  silver  were  used  in  its  construction. 

Assume  that  the  volume  of  an  alloy  is  the  combined  volumes  of  the 
components,  and  take  the  density  of  gold  as  19.3  and  that  of  silver  as  10.5. 

4.  A  10-gm.  weight  placed  upon  a  block  of  wood  weighing  30 
gm.  sinks  it  to  a  certain  point  in  water.     In  a  salt  solution  it 
requires  15  gm.  to  sink  the  wood  to  the  same  point.     Find  the 
density  of  the  salt  solution. 

5.  If  the  density  of  ice  is  .918  and  that  of  sea  water  is  1.03, 
find  what    fraction  of  the  total  volume  of  an  iceberg  is  above 
water. 

6.  A  cylinder  of  cork  10  cm.  high  and  of  density  .2  floats  upon 
water.     If  the  air  above  the  water  be  removed,  will  the  cork  sink 
or  rise  in  the  liquid?     How  much? 

Assume  incornpressibility  in  both  cork  and  water. 

7.  Suppose  that  a    constant-weight    hydrometer  which  it  is 
desired  to  calibrate  is  immersed  in  two  liquids  whose  densities  are 
known  to  be  1.  and  1.1,  that  the  two  points  of  immersion  are 
accurately  marked,  and  that  the  intervening  stem  is  then  divided 
into  10  equal  parts.     Assuming  that  the  stem  is  accurately  cylin- 
drical, will  this  hydrometer  give  correct  readings  in  liquids  of 
intermediate  densities?     Why? 


XXI 


CAPILLARITY 

Theory 

One   of   the   fundamental   assumptions   made   in    elementary 

hydrostatics  is  that  a  liquid,  like  so  much  sand,  exerts  pressure 

merely  by  virtue  of  its  weight,  and  by  virtue  of  the 

Ordinary  J       J  J 

law  of  liquid   property,  which  it  possesses  in  common  with  all  fluids, 

of  transmitting  pressure  in  all  directions.*     Thus  if 

p0  be  the  pressure  (force  in  grams  per  unit  area)  exerted  upon  the 

surface  of  a  liquid  of  density  d,  then  the  number  of  grams  of 


*This  property  follows  at  once  from  the  fact  of  fluidity  and  the  fun- 
damental laws  of  mechanics.  Thus  let  A  (Fig.  96)  be  a  substance  concern- 
ing which  the  one  assumption  is  made  that  it  is  capable  of 
adjusting  itself  with  perfect  ease  to  any  change  in  the  shape 
of  the  containing  vessel.  Let /and/'  be  two  forces  acting 
upon  frictionless  pistons  1  and  2  of  areas  a  and  a'  respectively.  It  is 
required  to  find  the  ratio  which  must  exist 
between  the  forces /and/',  in  the  condition 
of  equilibrium.  Let  piston  1  move  uniformly 
down  a  distance  s  thus  crowding  out  of  cyl- 
inder 1  a  volume  of  fluid  as.  /'  must  then 
move  uniformly  up  a  distance  s'  such  that 
as  —  a '  s ' .  But  from  the  '  'principle  of  work" 
(scholium  to  Third  Law)  in  the  condition  of 
equilibrium  (rest  or  uniform  motion)  fs  = 

f's'.     Hence  -p-=  — r.  i-  e-.  the  forces  which 
must   act  on  the  pistons    in  the    condition  FIGURE  96 

of   equilibrium  are  directly  proportional  to 

their  areas.  Since  the  directions  of  the  forces /and/'  are  wholly  arbi- 
trary, there  results  the  law  first  announced  in  1653  by  the  French 
philosopher,  mathematician,  and  man  of  letters,  Pascal, — "The  forces 
transmitted  by  fluids  act  equally  in  all  directions  and  are  proportional  to 
the  areas  of  the  surfaces  upon  which  they  act," — a  law  which  finds  its 
most  beautiful  experimental  demonstration  in  the  hyrdaulic  press. 

181 


182 


MOLECULAR    PHYSICS    AND    HEAT 


Capillary 
phenomena. 


pressure  P,  which  exists  at  any  depth  z  beneath  the  surface,  is 
given  by  (see  Fig.  97) 

P=pQ  +  zd.  (160) 

There  follows  at  once  then  the  result,  in  general  confirmed  by 
experiment,  that  a  liquid  contained  in  a  series  of 
communicating  vessels  must  take  the  same  level  in 
all  of  them,  no  matter  how  different  they  may  be  in 
size  or  shape. 

But  it  was  observed  as  early  as  1500  by  that  most 
universal  of  all  geniuses,  Leonardo  da  Vinci,  that 
when  a  tube  approaches  capillary  dimen- 
sions water  rises  in  it  far  above  the  level 
in  the  outside  vessel  (see  Fig.  101).  Later 
and  more  careful  investigation  has  shown  the  existence 
FIGURE  97  of  a  large  number  of  different  phenomena  to  which 
the  ordinary  laws  of  hydrostatics  do  not  apply.  These 
are  usually  all  called  "capillary  phenomena,"  because  they  were 
first  observed  in  connection  with  capillary  tubes.  They  are  all 
manifestations  of  those  intermolecular  forces  which  were  assumed 
in  Ex.  XVIII  in  order  to  reconcile  the  existence  of  liquid  surfaces 
with  the  theory  of  molecular  motion.  This  section  is  devoted 
to  a  study  of  the  effects  of  these  forces ;  and  since  these  effects 
would  of  necessity  be  just  the  same  whether  the  molecules  are 
at  rest  or  in  motion,  the  fact  of  motion  will  for  the  present  be 
disregarded. 

The  simplest  experiments  place  the  existence  of  these  inter- 
molecular  forces  beyond  the  possibility  of.  doubt,  and  show  at  the 
same  time  that,  while  they  have  enormous  values  at 
short  range,  they  diminish  so  rapidly  with  the  distance 
as  to  become  wholly  inappreciable  at  distances  which 
still  amount  to  extremely  minute  fractions  of  a  milli- 
meter. Thus  a  drop  of  mercury,  instead  of  spreading  out  into  an 
infinitely  thin  layer,  as  it  would  do  if  gravity  alone  acted  upon  its 
molecules,  is  held  together  in  globular  form.  A  sheet  of  glass 
may  be  brought  extremely  close  to  a  surface  of  water  without 
appearing  to  be  attracted  toward  it  in  the  slightest  degree,  but  as 
soon  as  contact  is  made  the  glass  clings  to  the  water  with  remark- 
able tenacity.  The  surface  of  two  metal  blocks  may  be  brought 
to  within  a  thousandth  of  an  inch  without  showing  any  appre- 


Evidence  as 
to  the  exist- 
ence and 
nature  of 
molecular 
forces. 


CAPILLARITY 


183 


pressure. 


ciable  attraction,  but  as  soon  as  they  are  brought  somewhat 
nearer,  as  by  pressure  or  welding,  it  requires  tons  of  weight 
to  pull  them  apart  again.  The  operation  of  the  aspirator 
pump  described  in  Ex.  XVIII  is  due  to  the  attraction  between 
the  air  about  the  orifice  o  and  the  outpouring  current  of 
water. 

Starting,'  then,  with  these  two  facts,  (1)  the  existence  of 
intermolecular  forces,  and  (2)  the  rapid  diminution  of  these 
Laplace's  f  orces  with  the  distance,  *  the  great  French  geometrician, 
Laplace,  first  developed,  about  1807,  a  theory  of  capil- 
lary  action.  His  reasoning  was  somewhat  as  follows: 
Let  r  represent  the  distance  within  which  one  molecule  attracts 
another  with  a  force  which  is  large  enough  to  deserve  consid- 
eration. Laplace  called  it  the  radius  of  influence  of  molecular 
force.  It  is,  of  course,  not  a  quantity  the  magnitude  of  which  is 
definitely  fixed,  but  it  probably  never  exceeds  .00005  mm. 

Now  a  molecule  m'  in  the  interior  of  a  liquid  is  indeed  acted 
upon  by  all  the  multitude  of  molecules  lying  within  a  sphere 
of  radius  r  (see  Fig.  98);  but,  by  virtue  of  symmetry,  the 
resultant  of  all  these  forces  is  evidently  zero  ;  so  that  m'  may 
be  treated  as 
though  it  were 
under  the  in- 
fluence of  no 
molecular  force 
whatever.  Not 
so,  however,  with 
any  molecule  m 
which  is  nearer 

to  the  surface  than  the  distance  r.  For  while  the  molecules  within 
the  space  cefd  exactly  neutralize  the  effects  of  the  molecules  within 
the  space  acdl,  the  downward  resultant  of  the  forces  of  all  the 


FIGURE  98 


*In  order  to  account  for  this  rapid  diminution  with  the  distance  it  is 
not  necessary  to  assume  that  these  intermolecular  forces  are  any  other 
than  those  concerned  in  ordinary  gravitation.  For  the  law  of  inverse 
squares  can  be  considered  to  hold  for  the  attractions  of  masses  of  finite 
volume  only  so  long  as  the  distance  between  the  nearest  points  of  the 
attracting  bodies  is  infinitely  large  in  comparison  with  the  distances 
between  the  molecules  of  the  bodies. 


184  MOLECULAR    PHYSICS    AND    HEAT 

molecules  in  efg  is  wholly  unbalanced.  This  force  continually 
urges  m  into  the  interior  of  the  liquid.  But  all  the  other  mole- 
cules in  the  same  horizontal  layer  with  m  are  urged  inward  with 
the  same  force  and  all  the  molecules  in  other  layers  which  are 
within  a  distance  r  of  the  surface  are  urged  in  with  other  forces. 
The  result  of  all  these  unbalanced  forces  acting  upon  all  the  mole- 
cules contained  in  the  surface  layer  of  thickness  r  (called  the 
active  layer)  must  be,  then,  an  interior  pressure  of  uncertain, 
perhaps  enormous,  magnitude.  It  has  been  estimated  for  water 
at  something  like  10,000  atmospheres,  but  it  has  never  been 
measured  directly  and  never  can  be.  For  since  a  liquid  is  always 
bounded  on  all  sides  by  a  surface,  this  molecular  pressure  usually 
balances  itself  and  therefore  cancels  out  in  hydrostatic  measure- 
ments. Hence  it  is  that  equation  (160),  which  leaves  the  existence 
of  molecular  pressure  altogether  out  of  account,  and  treats  the 
liquid  molecules  as  though  they  were  so  many  independent  grains 
of  sand,  nevertheless  gives,  in  general,  correct  results.  But  it 
is  exactly  such  apparent  violations  of  the  ordinary  hydrostatic 
laws  as  are  shown  in  capillary  phenomena,  which  furnish  a  beau- 
tiful proof  of  the  existence  of  Laplace's  molecular  pressure. 

For  it  is  easy  to  show  that    this  pressure  must  be  greater 

underneath  a  convex,  and  less  underneath  a  concave  surface,  than 

it  is  beneath  a  flat  one.    Thus,  let  M  (Fig.  99)  represent 

Variation  '    .  r 

of  molecular   a  molecule  which  lies  in  the  active  layer  at  a  given  dis- 
tance  beneath  a  surface,  and  let  the  circle  drawn  about 


M  represent  the  sphere  of  influence  of  molecular 
forces.  The  surface  will  first  be  assumed  to  be  flat  (acb),  then 
convex  (ecf),  and  then  concave  (gcli).  In  the  first  case,  since 
pidjq  neutralizes  pacbq,  the  resultant  downward  force  acting 
upon  Mis  due  to  the  attraction  of  the  molecules  lying  within 
the  segment  iojd  of  the  sphere.  In  the 
second  case  plcdlq  neutralizes  pecfq,  and 
the  resultant  downward  force  is  due  to 
kold,  a  volume  which  is  greater  than 
iojd.  In  the  third  case  the  resultant  force 
is  due  to  mond,  a  volume  which  is  less 
than  iojd.  Hence  the  resultant  downward 
force  upon  the  molecules  in  the  active  layer 
FIGURE  99  is  greatest  beneath  the  convex,  and  least 


CAPILLARITY 


185 


beneath  the  concave,  surface.  It  is  evident  also  from  the  same 
kind  of  reasoning  that  the  greater  the  convexity,  the  greater  this 
pressure.  Thus,  if  the  pressure  beneath  a  plane  surface  be  repre- 
sented by  PQ  (Laplace  named  this  the  normal  pressure),  that 
beneath  a  curved  surface  is  P0  ±  p,  in  which  the  magnitude  of 
p  depends  upon  the  nature  of  the  liquid  and  the  magnitude  of 
the  curvature,  while  its  sign  is  +  or  —  according  as  the  surface  is 
convex  or  concave.  Laplace  proved  by  a  mathematical  analysis 
of  the  forces  exerted  by  segments  of  the  kind  shown  in  Fig.  99, 
that  p,  expressed  in  terms  of  a  characteristic  constant  A  of  the 
liquid  (called  its  capillary  constant),  and  the  two  principal  radii 
of  curvature  R  and  R'  of  the  surface,  is 


(161) 


This  makes  the  ascension  or  depression  of  liquids  in  capillary 
tubes  perfectly  intelligible.  For,  starting  with  the  fact  of  obser- 
va^011  (accounted  for  below)  that  a  liquid  in  a  small 
*u^e  assnmes  a  curved  instead  of  a  flat  surface,  its  rise 
depression.  or  fa;Q  jn  the  tube,  according  as  the  surface  is  concave 
or  convex,  follows  as  a  matter  of  course  from  a  very  simple  con- 
sideration of  the  pressures  involved. 

Thus,  the  correct  value  of  the  pressure  at  a  distance  z  below  a 
plane    surface   is    not  pQ  +  zd,  as  assumed  in   (160),  but  rather 
p0  +  P0  +  zd,    and  the  pres- 
sure at   the  same  distance 
z  beneath  a  concave  surface 
in  a  capillary  tube  (see  Fig. 
100)  is 


V^ 

>! 

•: 

)o 
**% 

}m 

~ 

I 

FI 

( 

;i 

r* 

HI 

3   100 

z 
>                     a 

FlGTJ 

RE  101 

Hence  from  Pascal's  Law  of 

the    equal    transmission  of 

pressure  (see  note,   p.  181) 

there  can  be  no  equilibrium 

until  the  stronger  molecular 

pressure  beneath  the  flat  surface  has   pushed  up   the  liquid  in 

the  tube  to  such  a  height  h  (see  Fig.  101)  that  the  total  pressures 


186  MOLECULAR    PHYSICS    AND    HEAT 

at  any  two  points  a  and  #  in  the  same  horizontal  plane  are  the 
same,  i.e.,  until 

•     pQ  +  PQ  +  Zd  =  PQ  +  \PQ  -  A    (-jj-  +  jj;  j       +  Z(l  +  lid  \ 


or 


R  and  R'  are  the  curvatures  at  the  point  considered,  e.g.,  1,  2, 
or  3  (Fig.  101),  and  h  is  the  elevation  of  this  point  above  the  out- 
side plane  surface. 

It  thus  appears  that,  correctly  speaking,  a  liquid  does  not 
rise  in  a  capillary  tube  because  of  a  capillary  attraction,  any  more 
than  it  rises  in  a  suction  pump  because  of  the  attraction  of  the 
vacuum  created  by  the  lifting  of  the  piston.  In  both  cases  the 
liquid  is  pushed  up  by  a  pressure  existing  outside.  In  the  case  of 
the  pump  this  is  the  atmospheric  pressure  acting  on  top  of  the 
water  in  the  cistern;  in  the  case  of  the  capillary  tube  it  is  the 
normal  pressure  P0  acting  in  the  surface  layer  of  the  outside 
liquid. 

If  it  were  possible  to  remove  entirely  the  molecular  pressure 

within  the  tube,  the  height  of  rise  would  be  a  measure  of  P0,  just 

as  the  height  of  rise  of  the  water  in  a  long  tube  from 

Measurement       ,.,,,..,.,  ,     . 

ofthecapti-    which  the  air  is  entirely  removed,  is  a  measure  of  the 

lar-y  constant.  1  0.  . 

atmospheric  pressure.  Since,  however,  nothing  more 
can  be  done  than  to  obtain  a  curved  surface  within  the  capillary 
tube,  it  is  only  the  capillary  constant  A  which  can  be  found  from 
observations  of  the  height  of  ascension  li,  the  density  d,  and  the 
radii  of  curvature  R  and  R'  [see  (162)]. 

In  the  general  case  it  would  be  difficult  to  measure  R  and  R ', 
but  if  the  tube  is  cylindrical,  then  it  follows  from  symmetry  that 
at  the  middle  of  the  meniscus  R  =  R ',  and  (162)  reduces  to 

9  A 

hd=^-  (163) 

If,  farther,  the  tube  is  so  small  that  the  height  of  the  meniscus  m 
(see  Fig.  101)  is  negligible  in  comparison  with  'h,  i.e.,  if  h  is 
practically  constant  for  all  points  of  the  surface,  then  it  follows 

from  (162)  that  the  curvature  l-^-  +  -™  )  is  also  practically  con- 
stant. But  the  only  surface  of  constant  curvature  which  can  ful- 


CAPILLARITY  187 

fill  the  condition  imposed  by  (163)  is  a  section  of  a  sphere.  If 
finally,  then,  the  liquid  can  be  made  to  wet  completely  the  interior 
of  the  tube,  so  that  its  angle  of  contact  a  with  the  walls  is  180°, 
then  the  meniscus  must  be  a  hemisphere,  and  the  radius  R  is 
simply  the  radius  of  the  tube. 

Equation  (163)  is  then  applicable  to  all  cases  for  which  these 
conditions  hold.*  It  shows  that  the  height  of  rise  h  is  inversely 
proportional  to  the  radius  of  the  capillary — a  law  discovered 
experimentally  by  an  Englishman  in  1718  and  called  after  him  the 
law  of  Jurin.  Equation  (163)  thus  makes  the  measurement  of  the 
capillary  constant  a  very  simple  matter  in  the  case  of  liquids 
which  wet  solids  of  which  capillary  tubes  can  be  made. 

Another  interesting  conclusion  which 
can  be  drawn  from  the  above  reasoning 
is  that,  since  in  equilibrium  the  height 
h  depends  only  upon  the  curvature  and 
the  density,  the  dimensions  of  the  capil- 
lary above  or  below  the  point  of  contact       __ 
have  no  effect  whatever  upon  the  phe- 
nomena.    Thus,  if  water  be  drawn  up  FIGURE  102 
into  tubes  of  such  different  shape  as  a 

and  #  (Fig.  102),  it  should  come  to  rest  in  the  descent  at  pre- 
cisely the  same  level  in  both.  This  conclusion  is  wholly  con- 
firmed by  experiment. 

It   only  remains  to  show  why  a  liquid  in   a   capillary  tube 
assumes    a  curved  surface — a  task  of   no  difficulty  when   it   is 
remembered  that  a  liquid  surface  can  be  in  equilibrium 
'contact  U  °f    on^y  wnen  it  is  perpendicular  to  the  resultant  force 
acting  upon  its  molecules.     This  property  follows  sim- 
ply from  the  fact  of  mobility  of  the  particles.     For,  if  the  force 
acting  upon  the  surface  molecules  had  any  component  parallel  to 

*If  the  height  of  the  meniscus  is  not  wholly  negligible  in  comparison 
with  h,  the  mean  value  of  h  can  be  obtained  by  adding  ^R  to  the  height 
of  the  lowest  point  of  the  meniscus.  For  the  volume  of  the  liquid  above 
this  lowest  point  is  the  volume  of  a  cylinder  of  radius  R  and  height  R, 
minus  the  volume  of  a  hemisphere  of  radius  R;  or,  irR3  —  %irR3  =  %irR3. 
This  volume  divided  by  the  area  of  the  base,  viz.,  -n-R2,  gives  the  mean 

Tf 

height,  viz.,  •_-.    Formula  (163)  thus  modified  holds  for  tubes  of  as  much 
o 

as  2  mm.  diameter. 


188 


MOLECULAR    PHYSICS    AND    HEAT 


the  surface,  the  molecules  would  move  over  the  surface  in  obedi- 
ence to  this  component,  i.e.,  equilibrium  would  not  exist.  If, 
then,  on  (Fig.  103)  represent  the  line  of  junction  of  a  liquid  with  a 

solid  wall,  /!  the  resultant  of  all  the 
forces  exerted  upon  the  molecules  at 
o  by  such  portion  of  the  liquid  as 
lies  within  the  molecular  range  when 
the  liquid  surface  is  assumed  hori- 
zontal, and  /2  the  resultant  of  the 
forces  exerted  upon  the  same  mole- 
cules by  the  molecules  of  the  wall 
which  lie  either  above  or  below  the 
horizontal  line  passing  through  o, 
FIGURE  103  then  three  cases  may  be  distinguished : 

(1)    That    in   which   /t  =  2/2.      In 

this  case,  as  appears  from  Fig.  103,  the  cohesion  of  the  liquid  is 
exactly  equal  to  twice  the  adhesion  of  the  solid  and  liquid,  'and  the 
final  resultant  R  is  parallel  with  the  wall.  Hence  equilibrium 
exists  in  the  condition  assumed,  i.e.,  the  angle  of  contact  a  is  90°. 

(2)  That  in  which  /j  >  2/2.      The  resultant  R  then  falls  to 
the   right  of   on.     Hence    equilibrium   can  not    exist  until    the 
surface  near  o  has  become  convex,  i.e.,  until  the  angle  of  con- 
tact a  has  become  acute.     This   is  the 

case  of  liquids  which  do  not  wet  the 
wall.  If  the  substances  be  mercury  and 
glass  (Fig.  104),  equilibrium  is  reached 
when  a  is  about  43°.  It  is  to  be  ob- 
served, in  general,  that  this  angle  a  must 
always  be  the  same  for  the  same  two 
substances.  For,  on  account  of  the 
extreme  minuteness  of  the  sphere  of 
influence,  the  weight  of  the  particles 
contained  within  it  is  wholly  negligible 
in  comparison  with  the  molecular  forces, 
i.e.,  it  is  simply  the  relation  between  these  molecular  forces 
which  determines  the  angle  of  contact. 

(3)  That  in  which  /i  <  2/2.     The  resultant  R  then  falls  to 
the  left  of  on  (see  Fig.  105).     Hence  equilibrium  can  not  exist 
until  the  surface  near  o  has  become  concave  and  the  angle  of  con- 


FlGUBB  104 


CAPILLARITY 


189 


tact  obtuse.     In  all  cases  in  which  the  liquid 

completely  wets  the  solid,  the  angle  of  contact 

is  necessarily  180°,  i.e.,  a  thin  film  of  the  liquid 

lies  flat  up  against  the  face  of  the  solid.     This 

is  evident  from  the  consideration  that  when  a 

partially  immersed  body  is  raised  from  a  liquid, 

the  angle  of  contact  can  not  remain  constant 

at  any  value  less  than  180°  unless  the  liquid 

retreats  down  the  side  of  the  body  as  rapidly 

as  the  body  rises,  i.e.,  unless  the  liquid  be  one 

which  does  not  wet  the  solid. 

The  law  of  transmission  of  pressure  by  liquids  easily  accounts 

for  this,  at  first  view,  somewhat  surprising  result.  For,  in 
accordance  with  this  law,  the  molecular  pressure  P' 
existing  because  of  adhesion  at  a  point  in  the  liquid 

close  to  the  limit  of  contact  (see,  Fig.  106)>  is  trans- 
mitted undiminished  in  a  direction  parallel  to  the  sur- 
face of  the  solid,  and  therefore  constitutes  a  force  pushing  out  the 

molecules  at  c.  The  only  oppos- 
ing force  acting  to  prevent  the 
limiting  molecules  from  moving 
up  along  the  surface  is  the  verti- 
cal component  of  the  attraction 
/  exerted  upon  these  molecules 
by  such  portion  of  the  liquid  as 
lies  within  the  sphere  of  in- 
fluence drawn  about  c.  Hence, 
unless  the  ratio  of  the  cohesion 
to  the  adhesion  exceeds  a  cer- 
tain limit,  &>  thin  film  of  the 
liquid  must  spread  out  indefi- 
nitely over  the  surface  of  the 

solid.  This  conclusion  is  not  surprising,  since  it  means  simply 
that  a  body  which  attracts  a  liquid  strongly  enough  will  draw 
every  particle  of  it  as  near  as  possible  to  itself. 

Thus  it  is  that  a  drop  of  water  spreads  out  indefinitely  over  a 
perfectly  clean  glass  or  mercury  surface,  that  a  drop  of  olive  oil 
spreads  over  water,  or,  in  general,  that  any  liquid  spreads  out  over 
any  perfectly  clean  surface  which  it  wets.  But  such  perfectly 


FIGURE  106 


190 


MOLECULAR    PHYSICS    AND    HEAT 


clean  surfaces  are  very  difficult  to  obtain,  and  that  on  account  of 
the  prevalence  of  this  very  phenomenon.  Thus,  the  least  drop  of 
oil  touching  a  mercury  or  a  glass  surface  spreads  over  it  very 
quickly  and  completely  changes  the  effect  of  adding  a  drop  of 
water.  However,  such  familiar  facts  as  the  creeping  of  salt  solu- 
tions over  battery  jars,  of  kerosene  over  lamps,  or  the  rapid 
spreading  of  oil  over  water,  attest  the  correctness  of  the  above 
conclusions.  Of  course,  when  but  a  drop  of  the  liquid  is  present 
a  limit  to  the  spreading  must  be  reached  when  the  liquid  attains  a 
thickness  of  the.  order  of  magnitude  of  the  diameter  of  the  mole- 
cule. Rayleigh  measured  oil  films  on  water  which  had  a  thickness 
of  but  .000002  mm.  The  diameter  of  an  oil  molecule  can  not, 
therefore,  be  more  than  this. 

Another  interesting  result  which  may  be  deduced  from 
Laplace's  theory  of  molecular  pressure  is  that,  in  general,  a  liquid 
must  behave  as  though  its  surface  were  a  stretched 
of  surface  elastic  membrane.  For,  since  every  molecule  in -the 
active  layer  is  always  being  urged  into  the  interior,  it 
follows  that  as  many  molecules  as  can  possibly  do  so  will  leave 
this  layer  and  pass  within.,  i.e.,  a  liquid,  like  a  distended  rubber 
balloon,  will  always  tend  to  draw  together  into  the  form  which 
has  the  smallest  possible  surface  for  a  given  volume.  Thus  it  is 
that  all  bodies  of  liquid  which  are  not  distorted  by  gravity  or 
other  outside  forces  always  assume  the  spherical  form,  e.g.,  a 
raindrop,  a  soap  bubble,  a  globule  of  oil  floating  beneath  the  sur- 
face of  a  liquid  of  the  same  density. 

It  follows  again,  from  the  tendency  to  assume  the  form  of 
smallest  surface,  that  a  liquid  film,  a  form  of  enormous  surface, 

must  exhibit  a  sensible  contractil- 
ity. Experiment  amply  supports 
the  conclusion.  Thus,  a  soap  bub- 
ble may  be  observed  to  begin  to 
draw  back  into  the 
bowl  of  the  pipe  as 
soon  as  the  blower 
removes  his  mouth. 

A  wet  loop  of  cotton  thread  laid  upon  a  soap 
film,  as  in  Fig.  107a,  is  snapped  out  at  once  into 
circular  form,  as  in  Fig.  107b,  as  soon  as  the  FIGURE  ios 


FIGURE  107 


CAPILLARITY  191 

film  within  the  loop  is  pricked  with  a  pin.  A  film  formed  in 
the  frame  abdc  (Fig.  108)  snaps  the  piece  ab  back  to  cd  as  soon 
as  the  stretching  force  F  is  removed. 

Further,  Laplace's  theory  leads  to  the  remarkable  conclusion 
that  the  contractility  of  liquid  films  is  wholly  independent  of 
Tension  of  their  thickness.  For  the  work  which  is  performed  by 
Sendenfo/  an^  aSerL^  which  is  increasing  the  surface  of  a  liquid 
thickness.  \  consists  solely  in  bringing  new  molecules  from  the 
interior  to  the  surface,  against  the  force  of  the  molecular  pressure. 
Similarly  the  contractility  of  the  film  when  the  stretching  force  is 
removed  is  nothing  but  a  manifestation  of  the  force  of  molecular 
pressure  drawing  back  molecules  into  the  interior.  Hence  the 
work  done  by  the  outside  agent  when  the  surface  is  increasing, 
or  by  the  molecular  pressure  when  it  is  decreasing,  is  simply  pro- 
portional to  the  increase  or  decrease  in  surface;  i.e.,  the  work 
required  to  pull  down  ab  (Fig.  108)  a  given  distance,  e.g.,  1  mm., 
must  always  be  the  same,  whether  the  film  has  been  stretched 
little  or  much,  i.e.,  whether  it  is  thick  or  thin.  It  is  because 
this  conclusion  is  at  variance  with  the  law  which  governs  the 
stretching  of  solids  (stretching  force  proportional  to  cross-section) 
that  it  appears  strange.  It  is,  however,  completely  confirmed  by 
experiment.  Thus  the  fact  that  the  loop  of  Fig.  107b  takes 
the  accurately  circular  form  shows  that  it  is  subjected  to  pre- 
cisely the  same  force  at  all  points  on  its  circumference;  yet 
the  varying  colors  of  the  film  show  that  it  has  a  widely  varying 
thickness. 

It  is  to  be  observed,  however,  that  this  conclusion  as  to 
the  constancy  of  F  should  hold  only  so  long  as  the  film  is 
more  than  twice  as  thick  as  the  active  layer;  for  after  this 
thickness  has  been  reached  the  molecular  pressure,  and  hence 
also  the  work  required  to  bring  a  new  molecule  from  the  middle  to 
the  surface,  must  begin  to  diminish.  It  is  probable,  however,  that 
the  film  must  break  at  this  point.  Hence  it  is  that  the  smallest 
thickness  which  a  soap  film  can  have  is  usually  taken  as  a  measure 
of  the  diameter  of  the  sphere  of  molecular  influence.  This 
quantity,  as  measured  by  Johonnott  at  the  Ryerson  Laboratory  in 
1897,  is  .000012  mm. 

It  follows  from  the  constancy  of  F  that  if  ab  (see  Fig.  108)  be 
pulled  down  a  distanced,  the  work  done  by  ^is  equal,  to  Fd. 


192  MOLECULAK    PHYSICS    AND   HEAT 

But  this  work  is  proportional  to  the  increase  in  surface.  If,  then, 
Relation  ^  represent  the  amount  of  work  which  must  be  done 
moiSar  against  the  molecular  pressure  in  order  to  bring  enough 
mrface6  and  m°lecules  into  the  active  layer  to  form  one  new  sq.  cm. 
tension.  of  surface?  then,  since  the  total  increase  in  surface  (con- 
sidering both  sides  of  the  film)  is  %ab'd,  it  follows  that 

Fd  =  2ab-d'T,    or    T=-—>  (164) 

But  %ab  is  simply  the  length  of  the  line  of  surface  to  which  the 
stretching  force  F  is  applied.  And  since  the  value  of  .F  depends 
not  at  all  upon  the  thickness  of  the  liquid,  but  only  upon  the 
length  of  the  attached  surface  line  2ab,  and  upon  a  quantity  T 
which  is  proportional  to  the  normal  molecular  pressure,  it  is 
obvious  that  it  is  merely  the  surface  of  the  liquid  down  to  a 
depth  r,  i.e.,  the  active  layer,  which  is  to  be  regarded  as  the  seat 
of  the  contractile  force  F  of  the  film.  Finally  then,  since  when 
%ab  =  1, 

T=  F,  (165) 

it  follows  that  Laplace^s  molecular  pressure  manifests  itself  in  any 
liquid  surface  as  a  tangential  contractile 
force  (see  Fig.  109)  which  is  numerically 
equal,  in  grams  per  cm.  of  length,  to  the 
quantity  of  work,  expressed  in  gm.  cm., 
FIGURE  109  which  is  required  to  bring  up  into  the  active 

layer,     against     the     molecular    pressure, 
enough  molecules  to  form  one  new  sq.  cm.  of  surface. 

A  rather  interesting  experiment  has  been  devised  to  illustrate 
this  fact  of  contractility  when  but 
one  surface  of  a  thick  film  is  al-  ' 
lowed  to  contract.    A  shallow  ves- 
sel with  one  side  cd  movable  about  c 
c  is  filled  with  water  (see  Fig.  110).                     FIGURE  no 
As  soon  as  the  thread  t  is  burned, 

the  side  cd  is  pulled  over  into  the  vessel,  in  spite  of  the  fact  that 
the  weight  of  the  liquid  would  tend  to  press  it  more  firmly  against 
the  support  e. 

Now  the  pressure  existing  within  a  rubber  balloon  may  be 


CAPILLARITY  193 

easily  calculated  from  a  knowledge  of  the  tension  in  its  elastic 
envelope.      Since,  then,  the  molecular  pressure  in  liquids  man- 

ifests itself   as  a   surface  contractility,  it  ought  to  be 
of  Laplace's    possible  to  obtain,  from  the  value  of  this  contractility, 

the     quantity   p    of     (161),     i.e.,    the    increase    in 
internal  pressure  which  is   due  to  a   curvature  of  the  surface. 
Thus  let  o-o-'  (Fig.  Ill)  represent  an  infinitely  small  rectangular 
element  of  a   convex  surface.      Let  the   arcs 
<r  and   a-'    correspond    to    the    two   principal 
radii    of    curvature  R  and  R  '.      If    the  ten- 
sion in  the  surface    has  a  force  of    T  grams 
per   unit  length,  then  the  number  of   grams 
of  force  F,  which  act  on  each  of  the  arcs  <r,  is 
To-.     These   forces   are,  of  course,  tangential 
to    the    surface    and     perpendicular    to     the 
arcs.       Similarly,     Ta     grams    of     force    act 
upon   each  of  the  arcs    </.       Since  the   sur-  FIGURE  111 

face  is  curved,  all  four  of  these  forces  have 
slight  components  in  the  direction  of  the  normal  on.  The 
pressure  beneath  the  element  is  evidently  the  sum  of  these 
normal  components  divided  by  the  area  o-o-'  of  the  element,  for 
pressure  is,  by  definition,  force  per  unit  area.  The  compo- 
nent "of  F  parallel  to  on  is  F  cosa  =  Fsin  ft.  But  in  the  limit 

sin  ft  =  ^57-  Hence  the  sum  of  the  normal  components  of  the  two 

fT*          f 

forces  F  is  —™—    Similarly,  the  sum  of  the  normal  components  of 

rj*i       ' 

the  two  forces  F'  is  —  ^—    Hence  the  pressure^  due  to  curvature 
is  given  by 

7W      7W 

<"•> 


That  this  formula  is  identical  with  that  denned  by  Laplace  by  a 
more  direct  but  much  more  difficult  method  attests  the  correctness 
of  the  reasoning,  and  also  shows  that  Laplace's  capillary  constant 
A  is  simply  the  tension  in  the  surface  per  unit  length  or  the 
amount  of  work  required  to  add  one  sq.  cm.  to  the  surface. 


194 


MOLECULAR    PHYSICS    AND    HEAT 


Object. 


Experiment 

To  compare  the  values  of  the  surface  tensions  of 
water  and  alcohol,  as  given  by  the  capillary  tube 
method,  with  the  results  obtained  by  measuring  directly  the  con- 
tractility of  films. 

DIRECTIONS. — 1.  Fill  two  small  beakers,  one  with  pure  dis- 
tilled water,  the  other  with  absolute  alcohol.     Then  prepare  a 
number  of  fresh  capillary  tubes  by  heating  to  softness 
kits  of  clean  glass  tubing  in  a  Bunsen  flame,  and  draw- 
ing them  down  to  diameters  of  from   .1   to   .5  mm. 
Select  several  tubes  which  seem  to  be  most  nearly  circular  in  form, 
and  attach  them  by  means  of  a  rubber  band  to  a  mirror  millimeter 
scale,  as  shown  in  Fig.  112.     Take  the  reading 
rQ  of  the  fixed  point  o  upon  the  scale,  by  placing 
the  eye  so  that  the  image  of  o  comes  into  coin- 
cidence with  o  itself.*     Then  immerse  the  lower 
ends   of  the   tubes  in   the  liquid  and  raise   and 
lower  the  clamp  c  several  times  (a  rack  and  pinion 
adjustment  is  to  be  preferred)  so  as  to  wet  thor- 
oughly the  capillaries  above  the  points  reached 
by  the  liquid.     Next  bring  o  exactly  into  contact, 
from  below,  with  the  liquid  surface,  slip  up  the 
capillary  alone  a  trifle,  and  take  the  reading  r  of 
the  bottom  of  the  meniscus  as  soon  as  the  level 
has  settled  back   to  its  position  of  equilibrium. 
It  is  clear  that  the  height  of  rise  li  is  given  by 
h  =  r  —  r0.     Mark  by  means  of   a  bit  of  wax  the 
point  to  which  the  liquid  rises  in  the  tube,  then  remove  it  from 
the    scale,    scratch  it   very   carefully  with  a    sharp    file  at  this 
point,  and  break  it  off  as  squarely  as  possible.     Stick 
™of the  tube  ^e  ^ro^en  tube  upright  against  the  side  of  a  block  of 
wood  by  means   of  soft  wax.     Then  focus  upon  the 
broken  end  a   microscope  which  is  provided  with  a  micrometer 
eyepiece.       Count  the  number  of  turnsf  and  fractions  of  a  turn 

*If  a  mirror-scale  is  not  available  the  eye  may  be  set  in  the  correct 
position  for  reading  upon  an  ordinary  scale,  by  pressing  a  small  piece  of 
mirror  glass  against  the  scale,  behind  the  point  o. 

fThe  counting  is  greatly  facilitated  by  means  of  the  toothed  edge 
which  is  found  in  the  field  of  view  of  the  eyepiece.  Each  tooth  cor- 
responds to  one  revolution. 


FIGURE  112 


CAPILLARITY 


195 


Calculation. 


Sources 
of  error 


increases 
a  rise  in 


which  must  be  given  to  the  micrometer  screw  in  order  to  cause 
the  movable  cross-hairs  to  traverse  exactly  the  internal  diameter 
of  the  tube.  Repeat  several  times,  using  in  each  case  a  different 
diameter.  Then  replace  the  capillary  tube  by  a  standard  mil- 
limeter scale,  and  find  in  the  same  way  the  number  of  turns  and 
fractions  of  a  turn  corresponding  to  1  mm.  From  the  two 
observations  find  the  diameter  D  of  the  tube  in  mm. 

In  (163)  lid  represents  a  pressure  expressed  in  grams  per  square 
centimeter.     Hence,  in  order  to  obtain  A  in  absolute  units,  the 
lid  of  (163)  must  be  multiplied  by  980,  and  both  R  and 
h  must  be  expressed  in  centimeters.     The  best  deter- 
minations have  given  for  water  at  15°,  A  =  75 ;  for  alcohol,  A  =  25.5. 
If  the  results  obtained  by  this  method  are  not  uniform,  it  will 
be  because,  on  account  of  the  presence  of  impurities,  the  wetting 
of  the  tube  is  not  perfect,  or  because  the  tube  has  not 
a  circular  section.     It  is  to  be  observed  also  that  A  is 
a  function  of  the  temperature,  diminishing  as  the  latter 
This  was  to  have  been  expected,   since 
temperature  corresponds  to    a  pushing 
apart  of  the  molecules. 

2.   In   order  to   make  a  direct  measurement  of 

the  surface  tension,  attach  a  very  light  wire  frame 

a  (Fig.  113)  to  a  delicate  helical  spring 

The  direct 

measurement  s,  and  by  means  of  an  elevating  table  #, 
raise  a  vessel  of  liquid  till  the  frame 
is  immersed.  Next  lower  the  table  carefully  by 
means  of  a  rack  and  pinion  r,  until  a  film  forms  be- 
tween the  prongs  of  the  frame.  Then  quickly  take 
the  reading  of  the  index  i  upon  the  mirror- scale 
m.  Before  repeating,  stir  the  liquid  vigorously  by 
means  of  a  glass  rod  which  has  been  carefully 
cleaned  in  a  Bunsen  flame.  Continue  this  operation 
until  a  number  of  consistent  readings  can  be 
obtained.  The  difference  between  this  reading 
and  that  taken  when  the  spring  and  frame  hang 
freely,  is,  of  course,  a  measure  of  the  force  of  ten- 
sion F  possessed  by  the  two  surfaces  of  the  film. 
In  order  to  reduce  this  force  to  dynes  observe  the  elongation  pro- 
duced by  a  known  weight  of  the  same  order  of  magnitude  as 


FIGURE  113 


196  MOLECULAR    PHYSICS    AND    HEAT 

F.     Then  apply  Hook's  Law  to  determine  F  accurately  in  grams. 

Finally  measure  the  distance  I  between  the  vertical  wires  of  the 

frame  a  with  an  ordinary  metric  scale  and  calculate  Tfrom  (164). 

Since  the  presence  of  the  least  particle  of  oil  upon  the  surface 

changes  completely  the  value  of  T,  it  is  of  great  importance  that 

the  frame  and  vessel  be  thoroughly  cleaned  with  caustic 

Precautions.  ,,  .  .    .      .  ,    ,, 

potash  beiore  the  experiment  is  begun,  and  that  care 
be  taken  not  to  touch  the  liquid  at  any  time  with  the  fingers. 
The  purpose  of  the  stirring  after  each  observation  is  to  break  up 
any  film  of  impurity  which  may  be  present  in  spite  of  all  precau- 
tions. It  will  usually  be  found  to  increase  the  reading  somewhat. 
The  readings  are  to  be  taken  only  when  a  distinct  film  is  visible 
between  the  prongs.  If  the  frame  continually  snaps  up  without 
forming  a  film,  clean  it  again  with  caustic  potash  and  lower  the 
table  more  slowly. 


Record 

1.  r0  =  —  No.  turns  of  microscope  micrometer  to  one  mm.  =  — 

Water  Alcohol  Density  =  — 

rl  (tube  1)  — rz  (tube  2)  =  —  r3  (tube  3)  =  -    -  r4  (tube  4)  =  - 

.:hl  =  -  h2  =  -  .".  ha  =  —  h4  =  - 

Dl  (in  screw  turns)= D2  =  — 


Mean  A[=T]=—                                    Mean  A[=T]  =  - 
2.  Rd'g  with  film  = zero  =  —  Rd'g  with  film  = zero  =  — 


Problems 

1.  The  gifted  American  physicist,  Joseph  Henry,  first  suggested 
in  1848  the  determination  of  the  capillary  constant  by  attaching  a 
manometer  to  a  soap  bubble,  and  thus  measuring  the  pressure 
existing  within  the  bubble.  Assuming  the  surface  tension  of  a 
soap  solution  to  be  70  dynes,  find  what  would  be  the  difference  in 
the  levels  in  the  arms  of  a  water  manometer  when  attached  to  a 
bubble  of  5  cm.  diameter. 


CAPILLARITY  197 

2.  Find  how  many  ergs  of  work  must  be  expended  to  blow  a 
soap  bubble  of  15  cm.  diameter. 

3.  A  drop  of  water  placed  in  a  conical 
tube   (see  Fig.    114)    is   observed  to  travel 
rapidly  toward  the  small  end;    but  a  drop 

of   mercury  travels  toward  the  large  end.  FIGURE  114 

Explain 

4.  How  high  will  water  rise  in  pores  which  are  .001  mm.  in 
diameter? 

5.  Deduce  formula  (163)  from  the  consideration   that,  in  a 
capillary  tube  in  which  the  angle  of  contact  is  180°,  the  total 
upward  force  is  the  surface  tension  acting  upon  a  line   whose 
length  is   the   circumference  of    the  tube,  while  the   balancing 
downward  force  is  the  weight  of  the  liquid  raised. 

6.  Explain  from  considerations  of  molecular  pressure,  how  a 
needle  or  any  small  body  which  is  much  more  dense  than  water, 
may   yet   float   upon   it  provided  a  <  90°.     Could  it  ever  float 
if  o>  90°? 


XXII 
CALORIMETRY 

Theory 

Calorimetry  is  that  branch  of  Physics  which  deals  with  the  meas- 
urement of  heat  quantity.      It  had  its  beginning  about  the  year 
1760  with  the  work  of  the  Scotch  chemist  and  physicist, 
J°sePn  Black,  the  originator  of  the  caloric  theory,  and 
the  ^rs^  investigator  to  draw  a  sharp  distinction  between 
heat  and  temperature;  in  fact  the  first  to  give  any  care- 
ful definition  of  the  term  heat. 

That  bodies  change  in  temperature  is  a  fact  of  direct  observa- 
tion, but  the  notion  that  a  something  called  heat  passes  between 
bodies  of  changing  temperature  is  of  the  nature  of  an  hypothesis. 
This  hypothesis  has  taken  two  forms.  With  Black  and  his  fol- 
lowers, the  so-called  calorists,  heat  was  an  imponderable  fluid,  the 
caloric,  the  passing  of  which  into  or  out  of  a  body  was  the  cause 
of  temperature  change.  The  unit  of  heat,  the  calorie,  was  then 
defined  as  the  amount  of  heat  which  must  enter  or  leave  1  gram  of 
water  in  order  to  produce  1  degree  of  change  in  its  temperature. 

With  Joule,  Clausius,  and  practically  all  physicists  of  the  latter 
half  of  the  nineteenth  century,  a  rise  in  temperature  represents  an 
increase,  not  in  the  quantity  of  a  contained  heat  fluid,  but  simply 
in  the  mean  kinetic  energy  of  the  molecules.  The  calorists'  defi- 
nition of  the  heat  unit  has,  however,  been  retained  in  its  original 
form,  their  concept  of  the  transfer  of  a  heat  fluid  being  simply 
replaced  by  the  concept  of  a  transfer  of  molecular  energy,  kinetic 
or  potential,  or  both.  A  knowledge  of  the  caloric  theory  is  now 
important  only  because  of  the  light  which  it  throws  upon  the 
terminology  of  heat.  The  theory  was  altogether  abandoned  after 
Joule's  demonstration  of  the  equivalence  of  heat  and  work. 

Up  to  Black's  time  it  was  generally  supposed  that  the  rise  in 
temperature  of  a  substance  in  contact  with  a  hot  body  was  con- 
tinuous; but  Black  pointed  out  that  while  ice  or  snow  is  changing 

198 


CALORIMETRY  199 

into  water  it  maintains,  if  well  stirred,  a  perfectly  constant  tem- 
perature, no  matter  how  hot  may  be  the  stove  with  which  it  is 
in   contact.     In   order   to   explain   this  phenomenon, 

Origin  of  the 

term  patent  together  with  the  inverse  one  that  the  condensation 
of  steam  or  the  freezing  of  water  is  accompanied  by  a 
large  evolution  of  heat,  Black  assumed  that  a  certain  amount  of 
the  caloric  always  became  hidden  or  latent  at  the  time  of  a  change 
from  the  solid  to  the  liquid,  or  from  the  liquid  to  the  gaseous 
condition.  For  example,  since  it  was  found  that  the  mixing  of  1 
gram  of  ice  at  0°  and  1  gram  of  water  at  80°  C.  would  yield  2 
grams  of  water  at  0°,  or.  that  2  grams  of  water  at  40°  was 
required  to  just  melt  1  gram  of  ice  at  0°,  80  calories  was  taken  to 
be  the  latent  lieat  of  fusion  of  ice. 

According  to  the  modern  mechanical  theory,  the  temperature 
of  a  substance  remains  constant  while  the  change  of  state  is  going 
on  simply  because  the  energy  of  motion  communicated 
V  latent  to  the  molecules  in  contact  with  the  hot  body  is  at 
once  transformed  into  energy  of  position ;  that  is,  the 
heated  molecules  immediately  break  away  from  the  forces  which 
have  been  holding  them  in  the  given  state  (solid  or  liquid,  as  the 
case  may  be),  and  thereby  lose  their  increased  velocities  (i.e., 
their  increased  temperature)  as  rapidly  as  they  receive  them.  The 
operation  is  wholly  analogous  to  that  in  which  a  body  shot  up 
from  the  earth  loses  its  velocity  in  raising  itself  against  gravity. 
Thus,  although  the  old  terms  of  the  calorists,  latent  heat  of  fusion 
and  latent  heat  of  vaporization,  are  still  retained,  these  latent  heats 
represent  to-day  only  given  changes  in  the  potential  energy  of  the 
molecules,  just  as  a  given  rise  in  temperature  represents  a  given 
change  in  their  mean  kinetic  energy. 

But  it  must  not  be  supposed  that  changes  in  the  kinetic  and 
potential  energies  of  the  molecules  may  not  take  place  simultane- 
ously. In  fact,  there  is  but  a  limited  number  of  substances,  those 
in  general  which  are  of  a  crystalline  structure,  which  show  at  any 
points  a  change  in  potential  energy  unaccompanied  by  a  change  in 
kinetic,  i.e.,  by  a  rise  in  temperature.  Thus  wax,  resin,  gutta 
percha,  glass,  alcohol,  carbon,  and  a  great  number  of  other  sub- 
stances pass  gradually  through  all  stages  of  viscosity  in  melting  or 
solidifying.  In  such  cases  the  temperature  changes  continually; 
i.e.,  there  is  no  definite  point  at  which  the  substance  may  be  said 


200  MOLECULAR    PHYSICS    AND    HEAT 

to  begin  to  melt.  On  the  other  hand,  every  increase  in  the  tem- 
perature of  a  solid  is  accompanied  by  a  certain  amount  of  expan- 
sion, and  hence  by^ome  increase  in  the  potential  as  well  as  the 
kinetic  energy  of  the  molecules. 

The  following  table  shows  the  values  of  the  latent  heats  of 
some  of  the  commoner  substances : 


Water          

Melting     Latent  heat 
point,         of  fusion 
°C.             (calories) 

0.               79.9 

Boiling     Latent  heat 
point,    of  vaporization 
°C.           (calories) 

100.              536. 
80.2             94. 
118.               97. 
357.               62. 
447.             362. 

Bsnzol 

5.3 

30. 
46. 

2.8 
9. 
21. 
6 

27. 

Acetic  acid. 

...'.  16.5 

Mercury 

-  39.5 

Sulphur 

114. 

Silver 

..     .  970. 

Lead  

328. 

Platinum.  . 

..1780. 

It  was  discovered  very  early  that  the  quantity  of  heat  given  up 
by  1  gram  of  water  in  falling  through  1  degree  would  raise  very 

different  weights  of  other  substances  through  one 
Specific  heat,  degree,  e.g.,  33  gm.  of  mercury,  10.5  gm.  of  copper, 

8.9  gm.  of  iron,  2.3  gm.  of  turpentine,  etc.  The 
calorists  explained  these  facts  by  the  assumption  that  different 
substances  possess  per  unit  weight  different  capacities  for  the  heat 
fluid.  The  heat  capacity  of  a  body  was  then  defined  as  the  number  of 
calories  required  to  raise  the  body  through  1  degree,  and  the  specific 
heat  of  a  substance,  as  the  number  of  calories  required  to  raise  1 
gram  of  that  substance  through  1  degree.  These  definitions  are 
still  retained  now  that  heat  is  regarded  as  molecular  energy;  but 
the  fact  that  different  amounts  of  this  energy  must  be  communi- 
cated to  gram  weights  of  different  substances  in  order  to  produce 
the  same  increase  in  temperature,  i.e.,  the  same  increase  in  the 
average  kinetic  energy  of  translation  of  the  molecules,  is  attrib- 
uted to 

(1)  The  differences  in  the  number  of  molecules  contained  in 
gram  weights  of  different  substances ;  and 

(2)  The  differences  in  the  internal  work  which  are  incidental 
to  an  increase  in  temperature.     By  internal  work  is  meant  the 
work  done  in  increasing  the  distances  between  the  molecules,  in 
augmenting  the  energy,  kinetic  or  potential,  of  the  atoms  within 


CALORIMETRY  201 

the  molecules,  or,  in  general,  any  work  other  than  that  repre- 
sented in  the  increase  in  the  kinetic  energy  of  translation  of  the 
molecules  themselves,  or  in  the  expansion  against  the  force 
of  atmospheric  pressure. 

The  first  element  can  be  easily  investigated,  for  if  this  were  the 
only  cause  of  difference  in  the  specific  heats  of  different  sub- 
stances, these  differences  would  disappear  in  a  com- 
^eafsular  parison  of  quantities  which  represent,  not  equal 
weights,  but  equal  numbers  of  molecules.  Such  quan- 
tities can  evidently  be  obtained  by  taking,  in  each  case,  a  number 
of  grams  which  is  equal  to  the  molecular  weight  of  the  substance. 
This  quantity  has  been  given  the  name  of  a  gram-molecule,  and 
the  number  of  calories  of  heat  required  to  raise  1  gram-molecule  of 
a  substance  through  1°  C.  is  called  its  molecular  specific  heat,  or 
simply  its  molecular  heat.  The  molecular  heat  is  evidently,  there- 
fore, simply  the  product  of  the  specific  heat  per  gram  and  the 
molecular  weight.  The  following  table  contains  a  comparison  of 
a  very  few  specific  heats  per  given  weight,  and  specific  heats  per 
given  number  of  molecules : 

SPECIFIC  MOLECULAR  MOLECULAR 
GASES  HEAT          WEIGHT          HEAT 

I  Oxygen  (O2) 2175  32.  6.95 

Nitrogen  (N2) 2435  28.  6.81 

I  Hydrogen  (II2) 3.4090  2.  6.82 

Hydrochloric  acid  (HC1) 1845  36.4  6.72 

Carbon  monoxide  (CO) 2450  28.  6.86 

I  Nitric  oxide  (NO) 2317  30.  6.95 

Nitrous  oxide  (N2O) 2262  44.  9.95 

Carbon  dioxide  (CO2) 2163  44.  9.52 

Water  vapor  (H2O) 4805  18.  8.64 

SOLIDS 
Potassium  (K2) 1655  78.2          12. 94 

Sodium  (Na2) 2934  46.  13.50 

3<!  Silver  (Ag2) 0570  216.  12.32 

Copper  (Cu2) 0952  126.8  12.08 

Mercury  (Hg2) 0332  400.  13.28 

f  Oxide  of  copper  (CuO) 1420  79.4          11.27 

4J  Oxide  of  nickel  (NiO) 1588  74.8          11.87 

I  Oxide  of  mercury  (HgO) 0518          216.  11.19 


202  MOLECULAR    PHYSICS    AND    HEAT 

SPECIFIC  MOLECULAR  MOLECULAR 
SOLIDS  HEAT         WEIGHT         HEAT 

f  Chloride  of  calcium  (CaCl2) 1642  111.  18.2 

5J  Chloride  of  zinc  (ZnCl2) 1362  136.2  18.6 

I  Chloride  of  barium  (BaCl2) 0896  208.  18.6 

f  Sulphate  of  lead  (PbSO4) 0872  303.  26.4 

6 \  Sulphate  of  barium  (BaSO4) 1128  233.  26.3 

I  Sulphate  of  calcium  (CaSO4) 1966  130.  26.7 

This  table  shows  that  many  of  •  the  differences  in  specific  heats 
do  in  fact  disappear  upon  comparison  of  equal  numbers  of  mole- 
cules. The  differences  which  are  still  left  must  be  attributed 
wholly  to  the  second  cause,  viz.,  differences  in  internal*  work 
because  of  differences  in  molecular  structure  or  molecular  attrac- 
tion. Since  an  extensive  series  of  observations,  of  which  the  above 
table  is  a  small  fraction,  has  shown  that,  in  general,  the  molecular 
heats  of  chemically  similar  substances,  the  molecules  of  which 
possess  the  same  number  of  atoms  (see  table) ,  are  nearly  the  same 
in  a  given  state  of  aggregation  (law  of  Neumann),  it  must  of  course 
be  inferred  that  the  internal  works  are  also  the  same  for  such 
similar  substances. 

Since  the  molecules  of  gases  are  not  subjected  to  appreciable 
mutual  attractions  it  might  be  expected  that  with  molecules  of 
equal  complexity  the  molecular  work  would  be  less  in  the  gaseous 
than  in  the  solid  or  liquid  condition  (see  groups  1  and  3).  Again, 
it  would  be  natural  to  conclude  that,  for  substances  in  the  same 
state  of  aggregation,  the  internal  work  would  in  general  increase 
with  the  complexity  of  the  molecule.  Both  of  these  inferences 
are  seen  to  be  in  accordance  with  the  facts  presented  in  the  table. 

The  law  of  Xeumannf  is  not  exact,  nor  could  it  be  expected  to 

*  The  external  work  done  in  expanding  against  atmospheric  pressure 
may  be  neglected  for  solids  and  liquids.  For  gases  it  is  a  constant  quan- 
tity. 

f  The  law  of  Neumann  is  an  extension  of  a  law  discovered  in  1818  by 

Dulong  and  Petit  in  accordance  with  which  the  atomic  heats  (products  of 

specific  heat  and  atomic  weight)  of  all  the  solid  elements  are 

heats™  nearly  the  same,  amounting  to  about   6.3  (see  group  3  of 

table).     This  law  was  extended  in  another  direction  in  1848 

by  Woestyn  who  found  that  the  molecular  heat  of  a  compound  is  often 

equal  to  the  sum  of  the  atomic  heats  of  the  elements  contained  in  the 

compound.     It  is  difficult  to  recognize  in  this  discovery  anything  more 

than  an  interesting  empirical  law. 


CALORIMETRY  203 

be  iii  view  of  the  differences  in  the  attractions  which  exist  between 

different  sorts  of  molecules.     In  fact,  experiment  has  shown  that 

the  specific  heat  of  a  given  substance  is  not  constant, 

Variation 

of  specific       but  that  it  in  general  increases  steadily  with  the  tem- 

heate  with  J 

thetemper-  perature.  Hence,  save  in  the  case  of  the  permanent 
gases,  the  quantities  given  in  the  table  are  only  to  be 
regarded  as  mean  specific  heats  between  certain  temperatures,  e.g., 
15°  and  100°.  The  rate  of  increase  is  fortunately  slight  for  water 
and  for  most  solids,  so  that  in  ordinary  work  at  moderate  tem- 
peratures, it  may  be  disregarded.  But  for  most  liquids  it  is  far  from 
negligible;  for  example,  the  specific  heat  of  alcohol  is  .54  at  0° 
and  .64  at  40°C.  The  fact  of  the  dependence  of  the  specific  heat 
of  water  upon  the  temperature  was  first  established  by  Regnault 
by  the  method  of  mixture.  Thus,  for  example,  it  was  found  that 
equal  quantities  of  water  at  different  temperatures  do  not,  when 
mixed,  yield  exactly  the  mean  temperature;  or,  again,  that  the 
fall  of  10  gm.  of  water  from  60°  to  30°  does  not  heat  a  given 
quantity  of  cold  water  quite  as  much  as  the  fall  of  5  gm.  from  90° 
to  30°.  The  first  exact  work  upon  the  nature  of  this  variation 
was,  however,  done  by  Professor  Rowland  of  Johns  Hopkins  in 
1879,  in  accordance  with  whose  results  the  specific  heat  of  water 
diminishes  by  about  1  per  cent  from  0°  to  29°,  and  then  increases 
again  slowly  to  100°. 

In  view  of  these  facts  it  has  been  found  necessary  to  define  the 
calorie  as  the  amount  of  heat  required  to  raise  1  gm.  of  water,  not 
through  1  degree,  but  from  15°  to  16°  C.  The  defini- 
tions  of  lieat  caPacitJ  an(i  of  specific  heat  remain 
unchanged,  but  the  quantity  usually  obtained  by  exper- 
iment is  not  the  specific  heat  at  a  given  temperature,  but  rather 
the  mean  specific  heat  between  two  specified  temperatures.  Thus, 
if  Q  represent  the  amount  of  heat  passing  into  or  out  of  a  mass  m 
while  it  changes  in  temperature  from  ^  to  t^  then 

r^—r  =  mean  heat  capacity  between  ti  and  £2;       (167) 

— 


and  —  -,-^-  —  7-r  =  mean  specific  heat  between  tv  and  t2.        (168) 
m  (1%  —  ?i) 

Measurement        There  are  three  methods  which  have  been  used  for 
the  measurement  of  specific  and  latent  heats.    These  are  : 


204  MOLECULAR    PHYSICS    AND    HEAT 

(1)  The  method  of  mixture, 

(2)  The  method  of  cooling, 

(3)  The  method  of  fusion  of  ice  or  condensation  of  steam. 
TJie  method  of  mixture  is  most  common,  most  simple,  and,  in 

many  cases,  most  accurate.  It  consists  in  mixing  known  weights 
of  bodies  of  different  temperatures,  observing  the 
qTrnSSe^  resulting  temperature  and  then  writing  out  an  equation 
which  contains  on  one  side  all  of  the  heat  quantities 
lost  by  the  cooling  bodies,  and  on  the  other,  all  of  the  heat  quantities 
gained  by  the  warming  bodies.  The  specific  heat,  or  the  latent 
heat,  sought  is  the  unknown  quantity  of  this  equation.  For 
example,  suppose  that  it  be  required  to  find  the  latent  heat  of 
steam  x  from  an  experiment  in  which  p  gm.  of  steam  at  100°  are 
condensed  in  M  gm.  of  water.  Let  t{  be  the  initial  temperature 
of  the  water  and  tf  the  final  temperature  attained  by  the  mixture. 
Then  the  number  of  calories  lost  by  the  steam  in  condensing  is  px ; 
that  lost  by  the  condensed  steam,  in  passing  from  its  temperature 
of  condensation,  100°,  down  to  tf,  is  p  (100  -  tf).  The  heat  gained 
by  the  water  is  M(tf—  t^).  Hence,  if  no  heat  went  into  the  vessel, 
the  thermometer,  the  stirrer,  or  the  atmosphere,  the  equating  of 
the  heat  losses  and  heat  gains  would  give 

px  +p  (100  -tf)  =  M(tf-  tt).  (169) 

But,  as  a  matter  of  fact,  the  calorimeter  gains  heat  as  well  as 
the  water.  If  its  heat  capacity  be  represented  by  (7,  the  number 
Calorimeter  °^  cal°ries  so  gained  is  G  (tf  —  tt).  Similarly,  if  C'  and 
'mometer  cor-  ^"  are  ^ne  heat  capacities  of  the  thermometer  and 
rections.  stirrer  respectively,  (0'  +  C")(tf  —  tt)  calories  go  into 
them.  These  terms  must  therefore  be  added  to  the  right  side  of 
(169).  The  quantities  (7,  C",  and  C"  are  determined  either  from 
direct  observation  (see  Experiment)  or  from  the  product  of  the 
known  weights  and  specific  heats. 

To  determine  the  number  of  calories  gained  by  the  atmosphere 
through  radiation  from  the  calorimeter  it  is  necessary  to  know  (1) 

the  mean  temperature  tm  of  the  water  during  the  exper- 
^orrection  inient,  (2)  the  mean  temperature  tj  of  the  surrounding 

atmosphere,  (3)  the  number  of  minutes  of  duration  JV 
of  the  experiment,  and  (4)  a  quantity  Tc  called  the  radiation  con- 
stant of  the  calorimeter.  This  constant  represents  the  number  of 


CALORIMETRT  205 

calories  which  will  pass  per  minute  into  the  atmosphere  from  the 
calorimeter  when  the  temperature  of  the  latter  is  1°  above  that  of 
the  surrounding  air.  Then  in  the  above  experiment  the  number 
of  calories  L  lost  by  radiation  is 

L  =  k(tm-tj)N.  (170) 

This  also  must  be  added  to  the  right  side  of  (169).  Its  sign  will 
of  course  be  negative  if  tm<.tj. 

tm  is  determined  by  averaging  the  temperature  readings  made  at 
intervals  of  15  seconds  throughout  the  experiment,  k  is  obtained 
from  an  observation  of  the  number  of  minutes  n  required  for  the 
temperature  of  the  water  to  fall,  e.g.,  .5°,  after  the  conclusion  of 
the  experiment.  If,  then,  d  be  the  mean  difference  in  temper- 
ature between  the  water  and  the  room  during  this  time  of  fall, 
since  the  total  number  of  calories  lost  during  the  n  minutes  is 
MX  .5,  the  number  of  calories  which  would  be  lost  per  minute  for 
a  difference  of  temperature  of  1°  is 


It  will  be  observed  that  in  this  method  of  calculation  it  is  assumed 
that  a  body  radiates  heat,  i.e.,  falls  in  temperature,  at  a  rate 
which  is  proportional  to  the  difference  between  its  temperature 
and  that  of  the  surrounding  atmosphere.  This  is  a  law  of  cooling 
which  was  announced  by  Newton,  —  a  law  which  is  not  even 
approximately  correct.  However,  where  the  radiation  correction 
is  small  and  the  quantity  (tm  —  tj)  not  more  than  5°,  the  incorrect- 
ness of  Newton's  law  will  not  generally  introduce  an  appreciable 
error  into  the  result. 

The  radiation  correction,  however,   can  never  be  determined 
with  great  certainty.     Hence  the  effort  is  always  made  to  reduce 
it  as  much  as  possible.     This  is  done  (1)  by  using  such 
large  quantities  of  water  that  the  temperature  change 
*s  sma^»  (^)  ^J  making  the  initial  temperature  about 
as  much  below  the  room  temperature  as  the  final  is 
above  it,  (3)  by  highly  polishing  the  outside  surface  of  the  calorim- 
eter so  that  it  radiates  very  slowly,  and  (4)  by  inclosing  it  in  a 
still  air  space  surrounded  by  constant  temperature  walls.     With 
the  form  of  calorimeter  shown  in  Fig.  115,  the  radiation  correc- 
tion is  ordinarily  negligible  unless  the  experiment  lasts  more  than 


206 


MOLECULAR    PHYSICS    AND    HEAT 


FIGURE  115 


2  minutes  or  unless  the  difference  of 
temperature  between  the  calorimeter 
H  and  the  water-jacket  J  exceeds  2°C. 
A  very  sensitive  thermometer  is  of 
course  necessary  for  the  accurate  meas- 
urement of  such  small  temperature 
changes. 

|)         The  above  discussion  applies  in  gen- 
eral to  the  method  of  mixture,  whether 
it  be  a  latent  or  a  specific 

The  tempera-    .  ,  .   ,     .  ,   .         ^,     . 

ture  of  the       heat  which  is  sought.     But 

hot,  body. 

m    the    latter    case    some 
special  device  must  often  be  employed 

for  obtaining  accurately  the  temperature  of  the  hot  body,  and 
for  transferring  it  quickly  to  the  calorimeter.  Solids  which  are 
not  acted  upon  by  water  are  placed  in  finely  divided  form,  in  a 
wire  net,  and  heated  to  a  tempera- 
ture indicated  by  a  thermometer  5 
in  an  inclined  air  tube  F  which  is 
surrounded  by  a  water  or  steam  bath 
P  (see  Fig.  116).  In  cases  in  which 
there  might  be  any  heat-producing 
action  between  the  water  and  the 
solid,  the  latter  is  first  inclosed  in 
some  thin-walled  vessel  of  known 
heat  capacity,  then  heated  in  F  and 
dropped  into  H  as  before.  In  the 
case  of  liquids,  no  special  heating 

device  is  necessary,  since  thermometers  can  be  plunged  directly 
into  them ;  but  such  liquids  as  can  not  be  brought  into  contact 
with  water  must  be  inclosed,  like  similar  solids,  in  thin-walled 
vessels  of  known  heat  capacity. 

The   second   method    of    determining    specific    heats   is   the 
method  of  cooling.     It  consists  in  comparing  the  times  required 

for  a  given  closed  vessel  to  cool,  in  air,  through 
oflcooiirw)d  a  given  number  of  degrees,  first  when  filled  with 

water  and  then  when  filled  with  the  substance  whose 
specific  heat  is  sought.  If  Z±  and  Zz  are  the  two  times  of 
cooling,  and  Cl  and  C9  the  two  corresponding  heat  capacities 


FIGURE  lie 


CALOBIMETRY  207 

of  the  vessel  and  contents  in  the  two  cases,  then  it  may  be  shown 
that 


For,  since  radiation  takes  place  from  the  surface  layers  only,  it  is 
clear  that,  with  given  outside  conditions,  a  given  surface,  at  a 
given  temperature,  must  always  lose  heat  at  the  same  rate,  no 
matter  what  substance  may  be  inclosed  by  the  walls.  Let,  then, 
q  denote  the  number  of  calories  which  passes  out  of  a  given  sur- 
face at  temperature  t  in  an  infinitely  short  element  of  time.  This 
loss  in  heat  '  will  be  associated  with  a  small  fall  in  temperature, 
viz.  81?  which  will  be  determined  by  [see  (167)] 

q  =  CA.  (173) 

If  now  the  heat  capacity  be  changed  from  C±  to  C2  by  a  change  in 
the  contents  of  the  vessel,  the  new  fall  in  temperature  82  in  the 
same  infinitely  short  interval  of  time  and  at  the  same  temperature 
t  will  be  determined  by 

q  =  C&.  (174) 

Hence,  from  (173)  and  (174), 


i.e.,  at  any  given  temperature  of  the  surface  the  two  changes  in 
temperature  during  the  same  small,  interval  of  time  are  inversely 
proportional  to  the  heat  capacities.  This  is  exactly  equivalent  to 
the  statement  that,  at  a  given  temperature,  the  two  intervals  of 
time  required  for  the  same  small  change  in  temperature  are  directly 
proportional  to  the  heat  capacities.  Since,  then,  the  times  required 
to  pass  through  each  small  element  of  the  scale,  e.g.,  from  60°  to 
59°,  are  proportional  to  the  heat  capacities,  the  total  times  required 
to  pass  through  any  interval  of  temperature  made  up  of  these 
small  elements  must  also  be  proportional  to  the  heat  capacities. 
It  is  to  be  observed  that  this  conclusion  involves  no  assumption 
whatever  regarding  the  nature  of  the  law  of  cooling,  or  regarding 
the  relation  between  the  roles  played  by  convection  cur- 

Cmditions  •      ,1  ,. 

in  which         rents  and  by  true  radiation  in  the  cooling  process.     It 

method  of 

cooling  is        rests  solely  upon  the  assumption  01  similarity  in  the 

outside  temperature  conditions  and  uniformity  of  tem- 

perature in  all  parts  of  the  cooling  body.     This  last  condition  is 


208 


MOLECULAR    PHYSICS    AND   HEAT 


difficult  to  fulfil  when  the  cooling  vessel  contains  solids.     Hence 
the  method  has  not  proved  satisfactory  for  the  determination  of 

the  specific  heats  of  such  substances ; 
but  for  liquids  it  has  been  found 
both  accurate  and  convenient. 

The  third  method  of  measuring 
specific  or  latent  heats  consists  in 
either    determining    the 

The  ice  and 

steam  cai-       amount  oi  ice  m  which  a 

orimeters. 

heated  body  will  melt 
while  its  temperature  is  falling  to 
0°C.,  or  finding  the  amount  of  steam 
m'  which  a  cold  body  will  condense 
while  its  temperature  is  rising  to 
100°C.  The  heat  given  up  by  the 
body  in  the  first  case  is  then  evi- 
FIGURE  117  dently  80m.  That  taken  up  in  the 

second  case  is  5'36m'. 

The  ice  calorimeter  is  as  old  as  Black,  but  the  modern  form  is 
due  to  Bunsen  (see  Fig.  117).  The  ice  cap  b  is  formed  in  the 
water  W  by  inserting  a  freezing  mixture  into  the  tube  D.  The 
point  to  which  the  mercury  M  rises  in  the  graduated  capillary 
tube  Tis  then  noted.  The  hot  substance  is  next  dropped  into  Z>, 
where  it  melts  a  certain  amount  of  ice.  The  movement  of  the 
mercury  in  T  to  the  right  because  of  the  contraction  due  to  change 
of  state  is  proportional  to  the  amount  of  ice  melted.  The  value 
in  calories  of  one  division  of  T  is  determined 
by  inserting  into  D  a  substance  of  known 
heat  capacity.  This  calorimeter  has  proved 
very  valuable  in  determining  the  specific 
heat  of  very  small  bodies. 

The  steam  calorimeter  (Fig.  118)  has 
proved  of  especial  value  only  in  the  determi- 
nation of  the  specific  heats  of  gases.  A 
light  metal  globe  £,  full  of  the  gas,  is  sus- 
pended from  the  arm  of  a  balance  within 
a  chamber  of  known  temperature  t.  When 
steam  is  suddenly  admitted  into  this  chamber  through  E  it  con- 
denses upon  the  globe  and  the  walls  until  their  temperature  reaches 


FIGURE  118 


CALORIMETRY 


209 


100°.  With  proper  precautions  to  prevent  the  loss  of  drops  of  con- 
densed steam  (see  pan  s),  the  increase  tri  in  the  weight  of  the  globe 
represents  the  amount  of  steam  which  must  be  condensed  in  order 
to  raise  the  temperature  of  globe  and  contained  gas  from  t°  to  100°. 
If  the  heat  capacity  (7  of  the  globe  is  known,  that  of  the  con- 
tained gas  C'  can  evidently  be  found  from  ((7  +  (?')(100  —  t)  = 
536w'.  Accurate  results  can  not  be  obtained  with  either  of  these 
latent  heat  calorimeters  without  the  use  of  greater  precautions 
than  can  be  taken  ordinarily  in  intermediate  laboratory  courses. 


Object. 


Experiment 

To  compare  the  method  of  cooling  and  the  method 
of   mixture  in  the  determination  of  the  specific  heat 
of  turpentine. 

DIRECTIONS. — 1.  By  means  of  the  trip  scales*  find  the  .weight 
wc  of  the  nickel-plated  brass  vessel  A  of  about  30  cc.  capacity  (see 
Fig.  119).     Then  fill  it  with 

Observations         ...          '  . 

upon  rates      DOi ling    water    and    weigh 

of  cooling.  .     8. 

again  (wh).  Subtract  and 
obtain  the  weight  of  the  water  ww. 
Treat  the  blackened  vessel  B  in  the 
same  way,  filling  it  with  the  same  num- 
ber of  grams  of  water.  Suspend  both 
vessels  from  the  wooden  cover  D  by 
means  of  corks  and  thermometers,  as 
shown  in  the  figure.  Attach  the  cyl- 
indrical brass  vessels  E  and  F  by 
means  of  the  catches  at  et  and  immerse 
in  a  large  pail  or  battery  jar  full  of 
water  at  about  the  room  temperature. 

Take  a  very  exact  reading  of  each  thermometer  about  once  a 
minute  while  the  temperatures  are  falling  from  71°  to  39 °C. 
The  hour,  minute,  and  second  of  each  reading  must  be  accurately 
taken.  Since  the  eye  can  not  be  upon  both  the  watch  and  the 
thermometer  at  the  same  time,  it  is  advisable  to  have  in  the  room 
some  audible  second  marker  with  which  the  watch  can  be  com- 
pared just  before  each  reading.  This  may  be  dispensed  with  if 
there  are' two  observers,  one  to  record  temperatures  and  the  other 


FIGURE  '119 


210  MOLECULAR    PHYSICS    AND    HEAT 

to  record  times.  In  this  case  the  first  observer  should  tap  sharply 
upon  the  table  at  the  instant  of  each  temperature  reading,  at  the 
same  time  calling  out  either  A  or  B,  while  the  second  observer 
records  in  the  A  or  B  column  the  hour,  minute,  and  second  of  each 
tap.  The  mean  temperature  tj  of  the  water-jacket  may  be  found 
from  observations  taken  at  the  beginning  and  end  of  the  cooling 
process.  .  v 

Next  pour  out  the  water  from  A  and  B  and  dry  them  thor- 
oughly; fill  to  the  same  level  as  before,  this  time  with  hot  tur- 
pentine; obtain  the  weights  (w'b),  and  by  subtraction  the 
corresponding  weights  of  the  turpentine  wt ;  replace  in  the  bath 
and  take  a  new  set  of  observations  upon  the  rate  of  cooling 
between  71°  and  39°. 

With  the  observed  values  of  the  times  and  temperatures,  plot 
upon  a  large  sheet  of  coordinate  paper  four  smooth  full-page 
curves,  using  times  as  abscissae  and  temperatures  as 
curves  of  ordinates.  In  so  doing,  choose  the  scale  of  temper- 
ature so  that  the  lowest  observed  temperature  is  repre- 
sented by  a  line  near  the  bottom  of  the  page,  the  highest  by  a  line 
near  the  top.  Choose  the  scale  of  times  so  that  the  time  of  begin- 
ning of  observations  upon  the  cooling  of  the  polished  vessel  when 
filled  with  water  is  represented  by  a  line  (the  zero  of  time  for  this 
case)  which  is  near  the  left  side  of  the  page,  while  the  time  of 
conclusion  of  observations  upon  this  case  is  represented  by  a  line 
near  the  right  side  of  the  page.  Plot  the  other  curves  upon  the 
same  sheet  to  the  same  scale,  the  zero  of  times  being  in  each  case 
ttie  time  of  beginning  of  observations. 

Now  read  oif  very  carefully  upon  the  four  smooth  curves  the 
four  times  included  between  any  two  temperature  lines,  e.g.,  70° 
and  40°,  and  thus   obtain   from  both  A   and  B  the 
Computa-  £ 

uon.  quantity  -^-    If,  then,  c  and  c'  represent  the  respective 

*i 
heat  capacities  of  the  empty  vessel  A  and  of  the  thermometer,  and  s, 

Z      C 

the  specific  heat  of  turpentine,  the  equation  — i  =  -^  becomes 

Z2      02 

(176) 


From  (176)  s  can  easily  be  obtained  as  soon  as  c  and  c'  have  been 
found. 


CALOKIMETKY  ,  211 

To  obtain  c,  multiply  ivc  by  the  specific  heat  of  brass  (.095).     To 
obtain  c',  fill  vessel  A  with  water  at  the  room  temperature,  immerse 
and  read  carefully  the  thermometer;    then  plunge  it 
°^  wa^er?  withdraw,  wipe  off  the  adhering  water  ; 


m™  meters.  read  quickly  and  again  instantly  immerse  in  A.  Stir 
and  note  the  rise  in  the  temperature  of  the  water.  If 
then  w'  is  the  weight  of  the  water,  tt  and  tf  its  initial  and  final 
temperatures,  and  t  the  initial  temperature  of  the  thermometer, 
then  [see  (167)] 


From  any  two  of  the  four  cooling  curves  test  the  incorrectness 

of  Newton's  law  of  cooling  as  follows:     If  zl  and  zz  are  the  times 

required  to  cool  from  71°  to  69°  and  from  41°  to  39° 

Newton's 

law  of  respectively,  and  if  ts  is  the  temperature  of  the  jacket, 

then  if  Newton's  law  were  correct  it  would  follow  that 

«-m&b&  /178x 

^~w~t; 

2.  To  find  the  mean  specific  heat  of  turpentine  between  70° 
and  40 °C.  by  the  method  of  mixture,  first  heat  the  water  in  the 
jacket  J  (Fig.  115)  to  about  37°  and  hold  it  constant 
at  that  temperature  by  the  gentle  application  of  heat 
and  by  such  stirring  as  is  found  necessary.  Find  the 
weight  Wc  of  the  nickel-plated  calorimeter  H,  including  the 
stirrer,  then  fill  half -full  of  water  and  weigh  again  ( Wb).  Repre- 
sent the  weight  of  the  water  alone  by  Ww.  Give  the  water  a 
temperature  which  is  about  3°  below  that  of  the  jacket  and  set  it 
in  place  within  the  jacket  as  in  Fig.  115.  From  the  specific  heat 
found  by  the  method  of  cooling,  calculate  about  how  many  grams 
of  turpentine  will  need  to  fall  from  70°  to  40°  in  order  to  raise 
the  water  in  the  calorimeter  from  34°  to  40°.  From  this  and  the 
density  of  turpentine  (.87)  estimate  very  roughly  to  what  height 
the  mixture  of  water  and  turpentine  will  raise  the  level  of  the 
liquid  in  the  calorimeter.  Then  heat  to  about  73°  a  half -liter  or 
more*  of  turpentine  in  a  dipper  provided  with  a  lip ;  stir  it  thor- 
oughly with  a  thermometer  (No.  1)  until  the  temperature  falls  to 
about  70°,  then  take  an  accurate  reading  t't  and  very  quickly  pour 

*  A  large  quantity  is  used  so  that  the  cooling  may  not  be  too  rapid. 


212  MOLECULAR    PHYSICS    AND    HEAT 

about  the  estimated  volume  into  the  water.  The  temperature  tt 
of  this  water  should  have  been  taken  but  an  instant  before  with 
the  aid  of  a  more  sensitive  thermometer  (Xo.  2). 

Cover  the  calorimeter  as  quickly  as  possible  after  the  mixing 
and  stir  very  thoroughly  with  the  wire-net  stirrer  R  (see  Fig. 
115),  keeping  it,  however,  always  below  the  surface.  Take  a  read- 
ing every  15  seconds  from  the  time  of  mixing  until  the  temper- 
ature has  passed  its  highest  point  and  begun  to  fall.  The  first 
few  of  these  readings  will  of  course  be  uncertain,  but  they  have  a 
very  small  influence  upon  the  result.  Record  the  highest  temper- 
ature reached  tf,  then  withdraw  the  thermometer,  shake  back  into 
the  calorimeter  the  drops  which  adhere,  and  take  the  weight  Wm 
of  the  calorimeter  and  contents,  including  stirrer.  By  subtrac- 
tion obtain  the  weight  Wt  of  the  turpentine. 

To  find  the  radiation  constant  pour  out  the  mixture  and  fill  the 
calorimeter  with  a  volume  of  water  about  equal  to  the  volume  of 
the  mixture.     Find  the  weight  M  of  the  water  alone, 
for  radio,-       then  raise  it  to  about  the  final  temperature  ^/,  replace 
it  in  the  jacket  and  note  the  time  n  required,  with  con- 
tinual stirring,  to  fall  .5°C.     From  this  and  the  temperature  of 
the  jacket,  compute  first  the  radiation  constant   (equation  171), 
then  the  number  of  calories  lost  by  radiation  (equation  170). 

Calculate  the  heat  capacities  of  the  calorimeter  and  stirrer  by 
multiplying  their  joint  weight  by  .095,  obtain  the  heat  capacity  of 
Correction.-?  ^ne  thermometer  either  by  the  method  employed  in  1 
*eteran£im~  or  ky  the  following  process :  Estimate  the  volume  in 
thermometer.  Cubic  centimeters  of  that  portion  of  the  thermometer 
which  is  immersed,  by  noting  the  rise  of  water  in  a  narrow  gradu- 
ate when  the  thermometer  is  sunk  in  it  up  to  the  point  to  which 
it  was  wet  in  the  experiment.  Multiply  this  by  the  specific  heat 
of  mercury  per  cubic  centimeter,  viz.  13.6  x  .033  =  .45.  It  is 
because  this  is  about  the  same  as  the  specific  heat  of  glass  per 
cubic  centimeter,  viz.  2.5  x  .19  that  the  thermometer  may  be 
treated  as  though  it  were  made  entirely  of  mercury. 


CALORIMETRY  213 

Record 
A  B 

Wc  = Wc  =  

Wb=  —  W'b  — •  W0= W'b  = 

Ww=  . '.   Wt= . '.  Wiv=  . '.   Wt=  


ti= 

c  =  —  c'  =  —  t  = tf  =  — 

. '.  sp.  h't  of  turpentine  =  —  . '.  sp.  h't  of  turpentine  =  — 

Z±mn-  M-tj_  *,_  4Q-t,_ 

z2  70-  tj~  z2~  70  — */~ 

2.  1st  trial  2d  trial 

Wc=  Wc  =  - 

.;WW  =  -  tf=-  Ww=-  tf=- 

Wm=-  t'i  =  -  Wm=-  t'i=- 

.:Wt=-  t'f= *  Wt=—  t'f= * 

Radiation  cons'  t  (171)  M       ===  —  ?i  =  —  d  =  —  .-.k  =  — 

Radiation  correct'n  (170)  tm=  —  tj= N=  —  .  •.  L  =  — 

Heat  capacity  of  cal'r  and  stirrer  =  —  of  thermoin'r  =  — 

.  •.  sp.  h't  of  turpentine  =  —  .  •.  sp.  h't  of  turpentine  = 

Problems 

1.  50  gm.  of  ice  are  dropped  into  a  brass  calorimeter  contain- 
ing 800  gm.  of  water  at  27 °C.     The  calorimeter  weighs  150  gm. 
Find  the  final  temperature. 

2.  A  50  kgm.  block  of  ice  fell  30  meters.     How  many  grams 
of  ice  were  melted  by  the  heat  generated  by  the  fall?     (4.2  x  107 
ergs  =  1  calorie.) 

3.  According  to  very  careful  determinations  made  by  Violle 
the  quantity  of  heat  required  to  raise  1  gm.  of  platinum  from 
0°  to  t°  is  given  for  all  temperatures  by  the  formula  Q  =  .0317^  + 
.OOOOOG^2.     Find  the  temperature  of  a  Bunsen  flame  if  a  20  gm. 
platinum  ball  dropped  from  the  flame  into  377  gm.  of  water  at  0° 
raised  the  temperature  of  the  water  2°. 

*  t'f  is  tf  reducecf  to  terms  of  thermometer  No.  1.  It  is  found  by  com- 
paring No.  1  and  No.  2  at  about  the  temperature  tf.  It  is  evident  that 
this  proceeding  diminishes  errors  due  to  imperfect  thermometers. 


214  MOLECULAR    PHYSICS   AND    HEAT 

4.  What  should  be  the  result  of  mixing  10  gm.  of  snow  at  0° 
with  10  gm.  of  water  at  35°C.? 

5.  The  globe  of  a  steam  calorimeter  is  made  of  brass,  weighs 
50  gm.  and  has  a  volume  of  1  liter.     It  contains  nitrogen  at  a 
pressure  of  2  atmospheres.     The  temperature  of  the  chamber  is 
10°.     What  will  be  the  increase  in  weight  upon  the  admission  of 
steam? 


XXIII 
EXPANSION 

Theory 

The  true  coefficient  of  expansion  c  of  any  body  at  any  temper- 
ature is  defined  as  the  ratio  between  the  volume  increase,  pro- 
duced by  an  infinitely  small  rise  in  temperature,  and 
the  volume  of  the  body  at  zero.     Thus  if  F0,   F<p  and 
F,2  represent  the  volumes  at  0°,  ti°  and  tz° ,  then,  if  ^  and  tz  are 
infinitely  close  together, 

c=  J^~^-  (179) 

For  hydrogen  this  quantity  is  the  same  for  all  temperatures 
simply  by  virtue  of  the  definition  of  temperature;  for  it  is  the 
expansion  of  hydrogen  which  has  been  made  the  meas- 
es constant      ure  of  temperature  change  (see  p.  126).     For  the  other 

for  gases.  ...  J   .          .    .  . A  J .          '         .   _        T 

gases  it  is  constant  by  virtue  of  the  law  of  G-ay-Lussac 
(see  p.  121).  Hence,  for  gases  which  follow  closely  this  law,  equa- 
tion (179)  gives  the  correct  definition  of  the  expansion  coefficient, 
even  when  ti  and  t2  represent  widely  different  temperatures.  For 
such  cases  the  volume  at  tf°,  viz.  Ft,  expressed  in  terms  of  the 
volume  at  0°,  viz.  F0,  is  [cf.  (179)  when  ^  =  0] 

F,=  V,(l  +  ct}.  (180) 

The  density  at  t°,  viz.  Dt,  in  terms  of  the  density  at  0°,  viz.  D0, 

"ivr*i  ^1^1 

is  found  by  substituting  in  (180)  the  relation  =- r—  =  Volume. 

This  gives 

n    _          °    „  /'1ft1'\ 

jL/ 1   —   ~  :  I  J.OJ.I 

l  +  ct 

That  c  can  not  be  constant  for  most  liquids  and  solids  is  evi- 
dent from  the  fact,  already  mentioned  on  p  .  123,  that  ther- 
mometers made  from  liquids  or  solids  by  dividing  the  increase  in 
volume  between  0°  and  100°  into  100  equal  parts,  do  not  in  gen- 

215 


216 


MOLECULAR    PHYSICS    AND    HEAT 


stant  for 

solids  and 
than  .2 


eral  agree  at  intermediate  temperatures  with  the  hydrogen  ther- 
mometer. For  most  solids,  however,  and  for  mercury,  the  depart- 
ures are  slight  for  temperatures  below  100°.  For 
example,  a  mercury  in  glass  thermometer  graduated  in 

,     .     ,.       ,     ,      ,.,„  ,  ,       , 

the  manner   just  indicated    diners  from   a    hydrogen 
thermometer  at  no  point  between  0°  and  100°  by  more 
Hence,  in  ordinary  work  with  these  substances,  at  ordi- 
nary temperatures,  c  is  usually  considered  constant,  and  equations 
(180)  and  (181)  are  applied  precisely  as  in  the  case  of  gases.     For 
high  temperatures,  however,  this  approximation  can  not  be  used. 
The  expansion  coefficients  of  most  liquids  other  than  mercury 
increase  rapidly  with  the  temperature.     For  example,  in  passing 
from  0°  to  40°  the  coefficient  of  turpentine  increases 
™tfflc<Sant  3.9%  ;  of  alcohol,  4.4%  ;  of  bisulphide  of  carbon,  4.9%  ; 
liquids^        of  ether,  6.6%.     Between  0°   and  4°   water  possesses 
the  peculiar  property,  which  is  also  shown  by  certain 
alloys,  of  contracting  as  the  temperature  rises,  i.e.,  it  has  a  nega- 
tive coefficient.     It  is  evident,  then,  that  in  general  if  t^  and  t% 
represent  widely  different  temperatures,  (179)  gives 
the  mean  coefficient  between  ti  and   tz,  rather  than 
the  true  coefficient  at  any  particular  temperature. 
It  is  to  be  observed  also  that  in  the  case  of  both 
solids  and  liquids  (not,  however,  in  the  case  of  gases), 
%  the  increase   in  volume   (  Vt2  —  Vtl)   [see 

(179)]  is  so  small,  even  when  t1  and  tz 
differ  from  each  other  by  as  much  as  100°, 
that  the  error  introduced  into  (179)  by  replacing 
V0  by  the  volume  at  any  ordinary  temperature  is 
less  than  the  necessary  observational  error  in  obtain- 
ing the  increase  (  Vh  —  Vt}). 

The  most  accurate  and  convenient  method  of 
studying  the  coefficients  of  liquids  at  various  tem- 
Method  of  peratures  is  to  fill  a  glass  vessel  having  a 
pamionofx'  large  bulb  and  a  capillary  neck  (see  Fig. 
liquid*.  120),  to  raise  the  temperature  from  tl 

to  tz  and  to  observe  the  corresponding  rise  I  of  the  liquid  in  the 
neck.  If  a  represent  the  area  of  a  cross-section  of  the  tube,  then 
the  apparent  increase  in  the  volume  of  the  liquid  is  al.  This 
apparent  increase  is,  however,  in  reality  the  differences  between 


V0  replace- 
able by  V. 


FIGURE  120 


EXPANSION  217 

the  expansions  of  the  glass  and  of  the  liquid;  for  the  increase  in 
the  capacity  of  the  bulb  is  precisely  the  same  as  would  be  the 
increase  in  the  volume  of  a  solid  glass  vessel  of  the  same  volume 
as  the  interior  of  the  bulb.  This  is  evident  from  the  consideration 
that  a  solid  bulb  may  be  conceived  as  made  up  of  a  series  of  con- 
centric hollow  bulbs,  each  of  which  expands  independently  of  all 
the  rest.  Hence  the  increase  in  the  interior  capacity  of  each  hol- 
low shell  is  the  same  as  the  increase  in  volume  of  the  solid  bulb 
which  fills  it.  Hence,  if  F0  represent  the  interior  volume  of  the 
bulb  at  0°  and  y,  the  expansion  coefficient  of  glass,  the  increase 
in  capacity  of  the  bulb,  viz.  (F,2  —  Vtl)  is,  by  (179),  equal  to 
"Po(4  —  t\)y-  Similarly,  if  c  be  the  coefficient  of  the  liquid,  the 
real  increase  in  its  volume  is  VQ(tz  —  t^c.  Since,  then,  the 
apparent  increase  al  is  the  difference  between  these  quantities, 
there  results 


(182) 


an  equation  which  shows  that  it  is  impossible  to  obtain  c  from  the 
measurement  of  $,  Z,  F0,  ti  and  t%,  unless  y  is  already  known. 

A  similar  condition  exists  with  respect  to  the  volume  coeffi- 
cients of  solids.  For  one  of  the  most  precise  methods  of  measur- 
Measurement  ing  this  quantity  consists  in  finding  the  densities  Dtl 
(/oefflcientime  and  A2  of  the  solid  at  t,°  and  £°  by  the  loss  of 
of  solids.  Weight  method  (see  Ex.  XX),  and  then  substituting  in 
the  equation  which  results  from  replacing  the  volumes  in  (179)  by 

.   ,.         Mass        m,  . 

the  relation  .=-  -  —    This  gives 

Density 


But  the  density  of  a  solid  can  not  be  found  by  the  loss  of  weight 
method  unless  the  density  of  the  liquid  in  which  it  is  immersed  is 
known.  If  the  expansion  coefficient  c  of  one  single  liquid  could 
be  accurately  investigated  by  a  method  which  was  independent  of 
the  expansion  of  glass  or  of  any  other  substance,  this  liquid  could 
be  used  for  determining  y  for  any  glass  bulb  [see  (182)],  and  this 
bulb  could  thereafter  be  used  for  investigating  the  coefficients  of 
all  other  liquids.  This  would  in  turn  render  the  method  of  (183) 
serviceable  for  determining  the  coefficients  of  solids. 


218 


MOLECULAR    PHYSICS    AND    HEAT 


B 


Absolute 
expansion 
of  mercury. 


E 


FIGURE  121 


The  liquid  which  has  been  chosen  for  this 
special  investigation  is  mercury.     The  method 
used  was  devised  early  in  the  last 
century  by  Dulong  and  Petit.    Reg- 
nault  afterward  repeated  the  deter- 
minations and  obtained  results  in  close  agree- 
ment with  those  of  the  earlier  investigators. 
The  principle  of  the  method  is  very  simple, 
though  its  application  is  less  so.     The  appa- 
ratus consists  essentially  of  two  tubes  A  B  and 
CE  (see  Fig.  121)  which  are  connected  at  the 
bottom  by  a  capillary  tube  BE.     Tube  AB  is 
surrounded  with  melting   ice  and  CE  with  a 
water-jacket  of  known  temperature  t.      The 
levels  at  A  and  C  must  adjust  themselves  so 
that  the  pressures  at  B  and  J?  are  equal;   but  the  pressure  at  B 
is  DJiQ  (see  Fig.  121),  and  that  at  E  is  Dtht,  D0  and  Dt  being  the 
densities  of  mercury  at  0°  and  t°  respectively.     Hence 

h9D9=htDt.  (184) 

It  follows  from  (184)  and  (181)   that  the  value  of  c  between  0° 
and  t°  is  given  by 

c  =  ^-°-  (185) 

Regnault  obtained  for  the  mean  coefficient  of  mercury  between  0° 
and  100°,  .0001815. 

The  volume  (or  cubical)  coefficient  which  has  been  thus  far 
discussed  is  the  only  expansion  coefficient  which  liquids  possess ; 
unear  but  for  solids  there  exists  also  a  linear  coefficient  which 


expansion         .      _    , 

of  solids.        is  defined  by 


(186) 


in  which  ?0,  ltv  and  ?,2  are  the  lengths  of  any  given  line  of  the 
body  at  0°,  ^°,  and  /2°  respectively.  It  is  this  coefficient  which 
is  usually  made  the  subject  of  measurement  in  the  case  of  solids; 
for  the  cubical  coefficient  c  can  be  deduced  from  it  by  means  of 
the  simple  relation 

c  =  3X.  (187) 


EXPANSION  219 

To  prove  that  this  relation  exists,  let  /,  T,  and  I"  be  the  three 
edges  of  a  rectangular  block  at  a  temperature  of  t° ' .  These  three 
Proof  that  lengths  may  be  expressed  in  terms  of  X  and  the  cor- 
responding  lengths  at  0°,  thus  [see  (186)], 


Multiplication  gives 

m"         J     J'     J"      f-l     i     O\/  _i_  Q\2/2  _l_  \8/3\  •  (1  ft£\ 
^Qt-Qt        n       \  -^-    ~T"      d'T'V       I™    O/V     6           I       /V    t'       I    4  1    XOOI 

or 

Ft  =  Fo  (1  +  3X£  +  3X2f  +  X3/3).  (189) 

But  since  X  is  always  very  small,  scarcely  ever  greater  than 
.00003,  the  terms  in  X2  and  X3  are  wholly  negligible  in  comparison 
with  the  term  in  X.  Hence 

Vt  =  Fo  (l  +  3X2f).  (190) 

Comparison  with  (180)  shows  that  c  =  3X.     Q.  E.  D. 

Linear  expansion    coefficients  have   for  the   most  part  [been 
determined  by  one  of  two  very  simple  methods.     The  first  was 
introduced  by  \      i 

Lavoisier  and 
coefficients.      Laplace  to- 
ward the  end 

of   the    eighteenth    cen-      f| 
tury.   It  consists  in  plac-      Wjj^*^^ 
ing  against  a  rigid  wall  FIGUKE  m 

(see  Fig.  122)  one  end  of 

the  bar  which  is  to  be  heated,  attaching  an  optical  lever  to  the 
other  end,  and  measuring  the  expansion  precisely  as  the  elonga- 
tion of  the  wire  was  measured  in  Ex.  VIII. 

The  other  method,  which  is  the  one  now  in  use  at  the  Inter- 
national Bureau  of  Weights  and  Measures,  consists  in  focusing 
two  fixed  microscopes  upon  fine  scratches  near  the  ends  of  the  bar 
to  be  investigated,  and  measuring  directly  with  the  aid  of  a  microm- 
eter eyepiece  and  a  comparison  scale,  the  elongation  produced  by 
a  given  rise  in  temperature  (see  Fig.  123).  Different  bars  of  the 
same  material  show  very  considerable  differences  in  expansion 
coefficients.  Hence,  in  general,  the  numbers  given  in  tables  as 


220  MOLECULAR    PHYSICS    AND    HEAT 

the  coefficients  of  solids  must  be  looked  upon  merely  as  mean 
values. 

Experiment 
To  determine  the  linear  coefficient  of  expansion  of  a 

Object.        ._    _.         . 

hollow  brass  tube. 

Place  the  tube  a   (see  Fig.  123)   in  the  non-conducting  hair- 
felt  covering  #,  mount  it  upon  supports  as  in  the  figure,  and  focus 

the  two  micrometer  microscopes  A  and  B  upon  fine 
preliminary    scratches  near  the  two  ends  of  a.     In  this  setting  see  to 

adjustments.  ,  . .  .  ,         ,         •     ,  . ,  '-, 

it  that  the  microscope  is  placed  so  that  tne  scratch 
appears  on  that  side  of  the  field  of  view  which  is  farthest  from  the 


middle  of  the  rod.  This  is  done  so  that  the  expansion  may  not 
cause  the  scratches  to  move  out  of  the  field.  It  will  be  observed 
also  that  it  takes  into  account  the  inversion  of  image  produced  by 
the  microscope.  Connect  the  rubber  tube  m  with  the  water  tap 
and  take  the  temperature  of  the  water  by  holding  a  thermometer 
in  the  stream  which  emerges  from  o.  Then  set  the  movable  cross- 
hairs of  the  microscopes  accurately  upon  the  scratches  and  take 
the  readings  upon  the  scales  in  the  micrometer  eyepieces. 

This  is  done  by  first  rotating  the  micrometer  screws  in  such  a 
direction  that  the  reading  upon  the   circular  head   continually 
increases,  i.e.,  passes  from  0  toward  100,  at  the  same 
microscope      time  observing  in  what  direction  the  movable  cross- 
hairs pass  over  the  field  of  view.     If  it  be  found  that 
they  move  from  right  to  left,  then  the  reading  of  the  scratch  is 
the  number  of  turns   (i.e.,  teeth)   and  fractions  of  a  turn   (see 
micrometer   head)   through    which   the    micrometer    screw    has 
turned  in  bringing  the  cross-hairs  from  the  extreme  right-hand 
tooth  up  to  its  present  position.     This  is  read  off  directly  upon 


EXPANSION  221 

the  toothed  scale  and  the  micrometer  head.  It  will  be  observed 
that  in  order  to  facilitate  the  counting  of  the  teeth  every  fifth 
notch  is  more  deeply  indented  than  the  rest.  If  the  cross-hairs 
had  been  observed  to  move  from  left  to  rignt,  the  extreme  left- 
hand  tooth  would  of  course  have  been  chosen  as  the  point  of 
reference.  Let  the  recorded  reading  be  a  mean  of  a  number  of 
•settings,  and  in  each  setting,  in  order  to  avoid  the  error  due  to 
the  play,  or  back-lash,  between  the  nut  and  the  screw,  bring  the 
cross-hairs  up  to  the  scratch  from  the  same  side. 

As  soon  as  the  two  readings  have  been  taken,  turn  off  the  cur- 
rent of  cold  water,  using  extreme  precautions  against  disturb- 
ing the  tube  #,  transfer  the  rubber  connecting  tube  from  the 

water  tap  to  a  steam  boiler,  and  allow  a  rapid  current 
Swtraerjfe  °^  steaDi  to  pass  through  a  until  expansion  ceases. 

Take  again  a  set  of  readings  at  both  en  Is.  Then  find 
the  reducing  factors  of  each  microscope  by  focusing  upon  a 
standard  scale  and  observing  the  number  of  turns  to  the.  mil- 
limeter. Reduce  the  observed  expansion  to  millimeters.  Measure 
the  distance  between  the  scratches  by  means  of  an  ordinary  meter 
stick.  Obtain  the  temperature  of  the  steam  from  the  barometer 
height  and  the  table  in  the  Appendix.  Then  compute  A.  [see 
(186)]. 

Record 
No.  of   screw  turns  to  mm.    in  microscope  A  =  —  in  B  =  - 

1st  trial  3d  trial 

Temper-  Reading  Temper-  Reading 

ature          in  A          in  B  ature         in  A  in  B 

Water  Water 

Steam  Steam 

Differences  Differences 

Diff's  in  mm.  Diff's  in  mm. 


.-.  Expansion  coef.  c=  .*.  Expansion  coef.  c  = 

Problems 

1.  In  compensated  pendulums  the  expansion  of  one  set  of  rods 
lowers  the  bob,  while  that  of  another  raises  it.     Suppose  the  first 
set  to  be  made  of  iron  and  to  have  a  total  length  of  90  cm.     If  the 
material  of  the  second  set  is  zinc,  what  must  be  their  total  length? 

2.  If  a  clock  which  has  an  uncompensated  pendulum  made  of 


222  MOLECULAR    PHYSICS    AND    HEAT 

brass  keeps  correct  time  at  15°,  how  many  seconds  will  it  lose  per 
day  at  25°? 

3.  A  surveyor's  steel  tape  which  is   10  meters  long  is  correct 
at  15°.     What  is  its  error  at  0°?   at  100°? 

4.  A  glass  bulb  of  10  cm.  capacity  is  exactly  full  of  mercury  at 
20°.     How  many  grams  of  mercury  will  run  out  if  it  is  heated 
to  100°? 

5.  A  barometer  provided  with  a  brass  scale  correct  at  15° 
read  735.65  mm.  on  a  day  on  which  the  temperature  was  25°.    By 
means  of  the  expansion  coefficients  of  brass  and  mercury,  find  the 
correct  barometer  height  at  0°  upon  the  day  in  question. 

6.  A  cubical  block  of  steel  30  cm.  on  an  edge  floats  on  mer- 
cury.    How  far  will  it  sink  when  the  temperature  rises  from  15° 
to  75°? 


o; ';,;_' 


APPENDIX 

THE    MICROMETER    CALIPEB 

In  the  micrometer  caliper  (see  Fig.  124)  the  divisions  upon  the 
scale  c  correspond  to  the  pitch  of  the  screw  s.  When  the  jaws  ab 
are  in  contact  the  edge  of  the  sleeve  d  coincides  with  the  zero  line 
of  the  scale  c,  and  the  zero  division  upon  the  graduated  edge  d 
coincides  with  the  line  ec.  Hence,  when  the  jaws  are  separated 
by  a  rotation  of  the  milled  head  h,  the  whole  number  of  screw 
turns  of  separation  is  read  off  directly  upon  the  scale  c,  while  the 


FIGURE  124 


fraction  of  a  screw  turn  is  given  by  the  graduation  upon  the 
edge  d.  For  example,  if  the  pitch  of  the  screw  is  .5  mm.,  and  if 
there  are  fifty  divisions  upon  the  circumference  of  d,  then  each  of 
these  divisions  corresponds  to  a  motion  of  .01  mm.  at  ~b.  By  esti- 
mating to  tenths  of  a  division  upon  d,  the  separation  of  the  jaws 
can  be  read  to  .001  mm. 

To  make  a  measurement,  place  the  object  between  the  jaws  ab 
and  turn  up  the  milled  head  li  until,  with  light  pressure  between 
thumb  and  finger,  the  head  slips  through  the  fingers  instead  of 
rotating  farther.  In  the  best  instruments  the  head  is  arranged  to 
slip  upon  the  screw  as  soon  as  a  certain  pressure  has  been  reached. 
If  this  arrangement  is  absent,  take  great  care  not  to  crowd  the 
screw.  Without  removing  the  object  from  ab  read  upon  the 
scales  c  and  d  the  separation  of  the  jaws  to  .001  mm.  Then 

223 


224 


APPENDIX 


remove  the  object,  close  the  jaws,  using  the  same  pressure  as 
before,  and  if  the  zero  of  d  does  not  coincide  exactly  with  the  line 
ec,  observe  the  difference  in  thousandths  millimeters,  and  correct 
the  first  reading  for  this  zero  error  of  the  caliper. 

THE    VERNIER    CALIPER 

A  vernier  is  an  auxiliary  sliding  scale  which  enables  an  observer 
to  increase  somewhat  his  accuracy  in  estimating  the  fractional 
portion  of  the  last  small  division  of  the  main  scale.  For  example, 
it  is  seen  at  once  from  the  main  scale  DE  (Fig.  125),  that  the 

length  of  the  rod 
AB  is  about  2.7 
units.  The  use  of 
the  auxiliary  scale 
A  C  removes  all 

uncertainty  as  to  this  tenths  place,  and  even  makes  possible 
an  estimate  in  hundredths  place.  For  AC  is  so  divided  that 
ten  of  its  divisions  are  exactly  equal  to  nine  divisions  on  DE. 
Hence,  if  the  zero  of  AC  were  back  in  coincidence  with  the 
mark  2  of  DE,  the  mark  1  of  A  C  would  be  just  one- tenth  of 
one  of  the  DE  divisions  behind  the  mark  3  on  DE.  Similarly, 
2  would  be  two  tenths  behind  4,  3  three  tenths  behind  5,  and  so 
on.  Hence,  if  the  vernier  scale  were  moved  up  so  as  to  bring  its  1 
mark  into  coincidence  with  the  mark  nearest  to  it  on  DE,  the 
distance  which  its  zero  would  move  up  beyond  2  would  be  one- 


E 

SCALE 

D 

113 

[IS 

ill 

no  is 

s    IT 

6 

5,    |4- 

I3J2I1   JO 

•  . 

ilo 

i 

8l         7l 

e; 

Sl         4| 

si 

2|        1 

Q^™'~'-"  -•••"^-       "  "'"j 

VERNIER 
FlGUKB 

125 

A 

B 

FIGURE 


tenth  of  a  unit.     If  it  were  moved  up  so  as  to  bring  the  5  into 
coincidence,  the  zero  of  A  C  would  have  moved  up  five  tenths 


APPENDIX  225 

units  beyond  2.  In  general,  then,  it  is  only  necessary  to  observe 
which  mark  on  the  vernier  A 0  is  in  coincidence  with  a  mark  on 
DE  in  order  to  know  how  many  tenths  the  zero  of  AC  has  moved 
up  beyond  the  last  division  passed  on  DE.  A  glance  at  Fig.  125 
shows  that  6  has  already  passed  coincidence,  while  7  has  not  quite 
reached  the  coincidence.  The  length  of  AB  is  then  2.6+.  The 
next  figure  beyond  the  6  is  nearer  to  7  than  to  6,  i.e.,  it  is  between 
2.65  and  2.70.  If  then  the  length  be  recorded  as  2.67,  the  only 
estimate  will  be  in  the  figure  7,  i.e.,  in  hundredths  instead  of  in 
tenths  place  as  at  first.  Thus  a  vernier  which  has  10  divisions  to 
9  divisions  on  the  main  scale  reads  directly  to  tenths  of  the  main 
scale  divisions.  In  the  same  way  a  vernier  which  has  25  divisions 
equal  to  24  main  scale  divisions  reads  directly  to  twenty-fifths  of 
the  main  scale  divisions.  If  the  ratio  is  50  to  49  the  vernier  reads 
to  fiftieths,  etc.  Fig.  126  shows  a  vernier  caliper  in  which  the 
vernier  has  25  divisions  to  24  half -millimeter  divisions  on  the  main 
scale.  It  therefore  reads  to  .02  mm.  The  body  is  placed  between 
the  jaws  a  and  #,  and  the  reading  taken  directly  upon  the  scale 
and  vernier. 


226 


APPENDIX 


TABLES 

Table  1 

COEFFICIENTS  OF  ELASTICITY 


SUBSTANCES 

VOLUME  ELAS- 
TICITY =  K 

SIMPLE  RIGIDITY 
=  n 

YOUNG'S  MODULUS 

Water 

2.2  X  1 
2.6  X  1 
4.1  X 
10.8  X 
18.4  X 
14.6  X 
9.6  X 
16.8  X 
5.5  X 

Oio 
U11 

Mercury  

6.6  X  1 
10.8  X 
21.4  X 
19.6  X 
13.5  X 
12.3  X 
6.5  X 

6n 

Glass  

2.4  X  1 
3.7  X 
8.2  X 
7.7  X 
5.3  X 
4.5  X 
2.5  X 

W 

Brass    drawn 

Steel 

Wrought  iron 

Cast  iron  

Copper  .  . 

Aluminium  

Table  2 

DENSITIES 

Solids 


Aluminium 2. 58 

Brass (about)      85 

Brick 2.1 

Copper 8.92 

Cork 0.24 

Diamond 3.52 

Glass  (common  crown) 2.6 

"      (flint) 3.0-6.3 

Gold 19.3 

Ice  at  0°  C. 0.91 

Iron,  c.ast 7.4 


Iron,  wrought. 

Lead 

Nickel 

Oak 

Pine 

Platinum 

Quartz 

Silver 

Sugar 

Tin 

Zinc  . . 


7.86 

11.3 

8.9 

0.8 

0.5 

21  50 

2.65 

10.53 

1.6 

7.29 

7.15 


Mean  density  of  earth  is  5.5270. 

Liquids 


Alcohol  at  20°  C 0.789 

Carbon  bisulphide 1.29 

Ethyl  ether  at  0°  C 0. 735 

Glycerine 1  26 

Turpentine 0.87 


Mercury 

Sulphuric  acid 

Hydrochloric  acid. 

Nitric  acid 

Olive  oil 


13.596 
1.85 
1.27 
1.56 
0.91 


Gases  at  0°  C.  and  76  Centimeters  of  Mercury  Pressure 


Air,  dry 0.001293 

Ammonia 0.000770 

Carbon  dioxide 0.001974 

Chlorine 0.003133 


Hydrogen 0.0000895 

Marsh  gas 0.000715 

Nitrogen 0.001257 

Oxygen 0.001430 


APPENDIX 


22? 


Table  3 

DENSITY  OF  DRY  Am  AT  TEMPERATURE  t  AND  PRESSURE  Hmm.  OF  MERCURY 


t 

#=720 

730 

740 

750 

763 

770 

DIFFERENCE: 
PER  mm. 

10° 

.001181 

.001198 

.001214 

.001231 

.001247 

.001263 

16 

11 

1177 

1194 

1210 

1226 

1243 

1259 

1     2 

12 

1173 

1189 

1206 

1222 

1238 

1255 

2     3 

13 

1169 

1185 

1202 

1218 

1234 

1250 

3     5 

14 

1165 

1181 

1197 

1214 

1230 

1246 

4     6 

15° 

.001161 

.001177 

.001193 

.001209 

.001225 

.001242 

5     8 

16 

1157 

1173 

1189 

1205 

1221 

1237 

6    10 

17 

1153 

1169 

1185 

1201 

1217 

1233 

7    11 

18 

1149 

1165 

1181 

1197 

1213 

1229 

8    13 

19 

1145 

1161 

1177 

1193 

1209 

1224 

9    14 

20° 

.001141 

.001157 

.001173 

.001189 

.001204 

.001220 

15 

21 

1137 

1153 

1169 

1185 

1200 

1216 

1     2 

22 

1133 

1149 

1165 

1181 

1196 

1212 

2     3 

23 

1130 

1145 

1161 

1177- 

1132 

1208 

3     4 

24 

1126 

1141 

1157 

1173 

1188- 

--  1204 

4     6- 

'  25° 

.001122 

.001138 

.001153 

.001169 

.001184 

.001200 

5     7 

26 

1118 

1134 

1149 

1165 

1180 

1196 

6     9 

27 

1114 

1130 

1145 

1161 

1176 

1192 

7    10 

28 

1110 

1126 

1142 

1157 

1172 

1188 

8    12 

29 

1107 

1122 

1138 

1153 

1169 

1184 

9    13 

30° 

.001103 

.001119 

.001134 

.001149 

.001165 

.001180 

Correction  for  Moisture  in  Above  Table 


Dew-point 

Subtract 

Dew-point 

Subtract 

Dew-point 

Subtract 

Dew-point 

Subtract 

—10° 
—  8 
—  6 

—  4 

—  2 

.000001 
.000002 
.000003 
.000002 
.000003 

0° 

fc 

+6 
+8 

.000003 
.000003 
.000004 
.000004 
.000005 

+10° 

+  13 
+14 
+  16 
+18 

.000006 
.000006 
.000007 
.000008 
.000009 

+20° 

+22 
+24 
+26 
+28 

.000010 
.000012 
.000013 
.000015 
.000016 

Table  4 
DENSITY  OF  WATER 


Table  5 

DENSITY  OF  MERCURY 


TEMP.  C° 

DENSITY 

TEMP.  C° 

DENSITY 

TEMP.  C° 

DENSITY 

0° 

0.999884 

13° 

0.999443 

0° 

13.596 

1 

0.999941 

14 

0.999312 

10 

13.572 

2 

0.999982 

is 

0.999173 

12 

13.567 

3 

1.000004 

16 

0.999015 

14 

13.562 

3.95 

1.000000 

17 

0.998854 

16 

13.557 

4 

1.000013 

18 

0.998667 

18 

13.552 

5 

1.000003 

19 

0.998473 

20 

13.547 

6 

0.999983 

20 

0.998272 

22 

13.542 

7 

0.999946 

22 

0.997839 

24 

13.537 

8 

0.999899 

24 

0.997380 

26 

13.532 

9 

0.999837 

26 

0.996879 

28 

13.528 

10 

0.999760 

28 

0.996344 

30 

13.523 

11 

0999668 

30 

0.995778 

32 

13.518 

12 

0.999562 

100 

0.958860 

34 

13.513 

228 


APPENDIX 


Table  6 
SATURATED  WATER  VAPOR 

Showing  pressure  P  (in  mm.  of  mercury)  and  density  I)  of  aqueous  vapor  saturated 
at  temperature  t;  or  showing  boiling  point  t  of -water  and  density  D  of  steam  cor- 
responding to  an  outside  pressure  P. 


t 

P 

D 

t 

P 

D 

t 

P 

I) 

—10 

2.2 

2.3X10-6 

30 

31.5 

30.1X10-6 

88.5 

496. 

—  9 

2.3 

2.5  " 

35 

41.8 

39.3  " 

89 

505. 

—  8 

2.5 

2.7  " 

40 

54.9 

50.9  " 

89.5 

515. 

7 

2.7 

2.9  " 

45 

71.4 

65.3  " 

90 

525. 

428.4X10-6 

-  6 

2.9 

3.2  " 

50 

92.0 

83.0  " 

90.5 

535. 

—  5 

3.2 

3.4  " 

55 

117.5 

104.6  " 

91 

545. 

—  4 

3.4 

3.7  " 

60 

148.8 

130.7  " 

91.5 

556. 

—  3 

3.7 

4.0  " 

65 

187.0 

162.1  " 

92 

566. 

2 

3.9 

4.2  " 

70 

233.1 

199.5  " 

92.5 

577. 

—  1 

4.2 

4.5  " 

71 

243.6 

93 

588. 

0 

4.6 

4.9  " 

72 

254.3 

93.5 

599.6 

1 

4.9 

5.2  " 

73 

265.4 

94 

610.6 

2 

5.3 

5.6  " 

74 

276.9 

94.5 

622.0 

3 

5.7 

6.0  " 

75 

288.8 

243.7  " 

95 

633.6 

511.1  " 

4 

Si 

6/4  " 

75.5 

294.9 

95.5 

645.4 

5 

6.5 

6.8  " 

76 

301.1 

96 

657.4 

6 

7.0 

7.3 

76.5 

307.4 

96.5 

669.5 

7 

7.5 

7.7  " 

77 

313.8 

97 

681.8 

8 

8.0 

8.2  " 

77.5 

320.4 

97.5 

694.2 

9 

8.5 

8.7  " 

78 

327.1 

98 

707.1 

10 

9.1 

9.3  " 

78.5 

333.8 

98.2 

712.3 

11 

9.8 

10.0  " 

79 

340.7 

98.4 

717.4 

12 

10.4 

10.6  " 

79.5 

347.7 

98.6 

722.6 

13 

11.1 

11,2  " 

80 

354.9 

295.9  " 

'98.8 

727.9 

14 

11.9 

12.0  " 

80.5 

362.1 

99 

733.2 

15 

12.7 

12.8  " 

81 

369.5 

99.2 

738.5 

16 

13.5 

13.o  " 

81.5 

377.0 

99.4 

743.8 

17 

14.4 

14.4  " 

82 

384.6 

99.6 

749.2 

18 

15.3 

15.2  " 

82.5 

392.4 

99.8 

754.7 

19 

16.3 

16.2  " 

83 

400.3 

100 

760.0 

606.2  " 

20 

174 

17.2  " 

83.5 

408.3 

100.2 

765.5 

21 

18.5 

18.2  " 

84 

416.5 

100.4 

771.0 

22 

19.6 

19.3  " 

84.5 

424.7 

100.6 

776.5 

23 

20.9 

20.4  " 

85 

433.2 

357.1  " 

100.8 

782.1 

24 

22.2 

21.6  " 

85.5 

441.7 

101 

787.7 

25 

23.5 

22.9.  " 

86 

450.5 

102 

816.0 

26 

25.0 

24.2  " 

86.5 

459.3 

103 

845.3 

27 

26.5 

25.6  " 

87 

468.3 

105 

906.4 

715.4  " 

28 

28.1 

27.0  " 

87.5 

477.4 

107 

971.1 

29 

29.7 

28.5  " 

88 

486.8 

110 

1075.4 

840.1  " 

APPENDIX 


229 


Table  7 
REDUCTION  OF  BAROMETRIC  HEIGHT  TO  0°  C. 

(The  table  corrections  represent  the  number  of  millimeters  to  be  subtracted  from 
the  observed  height  h.  They  are  obtained  from  the  formula  (.000181  —  .000019)^,  the 
first  number  being  the  cubical  expansion  coefficient  of  mercury,  the  second  the  linear 
coefficient  of  brass.) 


t 

OBSERVED  HEIGHT  IN  mm. 

680 

690 

700 

710 

720 

730 

740 

750 

760 

770 

mm. 

mm. 

s  mm. 

mm. 

mm. 

mm. 

mm. 

mm. 

mm. 

mm. 

10° 

1.10 

1.12 

1.13 

1.15 

1.17 

1.18 

1.20 

1.22 

1.23 

1.25 

11 

1.21 

1.23 

1.25 

1.27 

1.28 

1.30 

1.32 

1.34 

1.35 

1.37 

12 

.32 

1.34 

1.36 

1.38 

1.40 

1.42 

1.44 

1.46 

1.48 

1.50 

13 

.43 

1.45 

1.47 

1.50 

1.52 

1.54 

1.56 

1.58 

1.60 

1.62 

14 

.54 

1.56 

1.59 

1.61 

1.63 

1.66 

1.68 

1.70 

1.72 

1.75 

15 

.65 

1.68 

1.70 

1.73 

1.75 

1.77 

1.80 

1.82 

1.85 

1.87 

16 

.76 

1.79 

1.81 

1.84 

1.87 

1.89 

1.92 

1.94 

1.97 

2.00 

17 

1.87 

1.90 

1.93 

1.96 

1.98 

2.01 

2.04 

2.07 

2.09 

2.12 

18 

1.98 

2.01 

2.04 

2.07 

2.10 

2.13 

2.16 

2.19 

2.22 

2.25 

19 

2.09 

2.12 

2.15 

2.19 

2.22 

2.25 

2.28 

2.31 

2.34 

2.37 

20 

2.20 

2.24 

2.27 

2.30 

2.33 

2.37 

2.40 

2.43 

2.46 

2.49 

21 

2.31 

2.35 

2.38 

2.42 

2.45 

2.48 

2.52 

2.55 

2.59 

2.62 

22 

2.42 

2.46 

2.49 

2.53 

2.57 

2.60 

2.64 

2.67 

2.71 

2.74 

23 

2.53 

2.57 

2.61 

2.65 

2.68 

2  72 

2.76 

2.79 

2.83 

2.87 

24 

2.64 

2.68 

2.72 

2.76 

2.80 

2.84 

2.88 

2.92 

2.95 

2.99 

25 

2.75 

2.79 

2.84 

2.88 

2.92 

2.96 

3.00 

3.04 

3.08 

3.12 

Table  8 

CAPILLARY  DEPRESSION  OF  MERCURY  IN  GLASS 


DIAM- 


HEIGHT  OF  THE  MENISCUS  IN  mm. 


ETEK 

&4 

0.6 

0.8 

1.0 

1.2 

1.4 

1.6 

1.8 

mm. 

mm. 

mm. 

mm. 

mm. 

mm. 

mm. 

mm. 

mm. 

4 

0.83 

1.22 

1.54 

'  1.98 

2.37 

5 

0.47 

0.65 

0.86 

1.19 

1.45 

1.80 

6 

0.27 

0.41 

0.56 

0.78 

0.98 

1.21 

1.43 

7 

0.18 

0.28 

0.40 

0.53 

0.67 

0.82 

0.97 

1.13 

8 

0.20 

0.29 

0.38 

0.46 

0.56 

0.65 

0.77 

9 

0.15 

0.21 

0.28 

0.33 

0.40 

0.46 

0.52 

10 

0.15 

0.20 

0.25 

0.29 

0.33 

0.37 

11 

0.10 

0.14 

0.18 

0.21 

0.24 

0.27 

12 

0.07 

0.10 

0.13 

0.15 

0.18 

0.19 

13 

0.04 

0.07 

0.10 

0.12 

0.13 

0.14 

230 


APPENDIX 


Table  9 

SURFA  CE  '  TENSIONS 
in  dynes  per  cm. 


Alcohol 

at  20° 

25  5 

Olive  Oil   . 

at  20° 

31  7 

Benzine  .  .    
Glycerine 

at  15° 

at  17° 

28.8 
63  1 

Petroleum  
Water  

at  20° 
at     0° 

23.9 
76  6 

Mercury 

at  20° 

4500 

Water  

at  20° 

740 

Table  10 

AVERAGE  SPECIFIC  HEATS 


Alcohol 

at  40° 

0  648 

Lead   

0°-100° 

0  0315 

Aluminium  .  . 

0°-100° 

0  2185 

Mercury  

20°-  50° 

0  0333 

Brass  

0.094 

Paraffine  

0  683 

Copper 

0°-100° 

0095 

Platinum 

0°-100° 

0  0323 

Germa  n  •  si  1  ver 

00946 

Silver 

0°-100° 

0  0568 

Glass 

0  20 

Steel 

0  118 

Gold 

0  0316 

Tin      .    . 

0°-100° 

0  0559 

Ice 

0  504 

Turpentine   .... 

0  467 

Iron  

0°-100° 

0.1130 

Zinc  

0  0935 

Table  11 

AVERAGE  COEFFICIENTS  OF  EXPANSION  BETWEEN  0°  AND  100°  C. 


Glass 


Cubical 
0.000025  |  Mercury....  0.0001815  |  Turpentine...  0.00105 


Aluminium..  0.000023 

Brass 0.000018 

Copper 0.000017 

Gold  .         . . .  0.000014 


Linear 

Iron  (soft)  . .  0.000012 

Iron  (cast)..  0.0000105 

Lead 0.000029 

Platinum.. .  0.000009 


Silver 0.000019 

Steel 0.000011 

Tin 0.000022 

Zinc 0.000029 


Table  12 
IMPORTANT  NUMBERS 

ic  =  3.1416.     7T2  =  9.8696.     £  =  0.31831.     Logarithm  IT  =  .49715. 

Base  of  the  natural  system  of  logarithms  e  =  2.7183. 

1  inch  =  25.4  millimeters.  1  meter  =  39.37  inches.  1  mile  ==  1.609  kilo- 
meters. 

1  kilogram  =  2.2  pounds.  1  ounce  =  28.35  grams.  1  grain  =  64.8  milli- 
grams. 

Mechanical  equivalent  of  1  calorie  (15°)  =  4.19  X  107  ergs. 


APPENDIX 


231 


NATURAL  SINES 


Angle 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

Complement 
Difference 

0° 

0.0000 

0017 

0035 

0052 

0070 

0087 

0105 

0122 

0140 

0157 

0175 

89° 

1 

0175 

0192 

0209 

0227 

0244 

0262 

0279 

0297 

0314 

0332 

0349 

88 

2 

0349 

0366 

0384 

0401 

0419 

0436 

0454 

0471 

0488 

0506 

0523 

87 

3 

0523 

0541 

0558 

0576 

0593 

0610 

0628 

0645 

0663 

0680 

0698 

86 

4 

0698 

0715 

0732 

0750 

0767 

0785 

0802 

0819 

0837 

0854 

0872 

85 

5 

0.0872 

0889 

0906 

0924 

0941 

0958 

0976 

0993 

1011 

1028 

1045 

84 

6 

1045 

1063 

1080 

1097 

1115 

1132 

1149 

1167 

1184 

1201 

1219 

83 

7 

1219 

1236 

1253 

1271 

1288 

1305 

1323 

1340 

1357 

1374 

1392 

82 

8 

1392 

1409 

1426 

1444 

1461 

1478 

1495 

1513 

1530 

1547 

1564 

81 

9 

1564 

1582 

1599 

1616 

1633 

1650 

1668 

1685 

1702 

1719 

1736 

80 

10 

0.1736 

1754 

1771 

1788 

1805 

1822 

1840 

1857 

1874 

1891 

1908 

79 

11 

1908 

1925 

1942 

1959 

1977 

199-1 

2011 

2028 

2045 

2062 

2079 

78 

12 

2079 

2096 

2113 

2130 

2147 

2164 

2181 

2198 

2215 

2233 

2250 

77  17 

13 

2250 

2267 

2284 

2300 

2317 

2334 

2351 

2368 

2385 

2402 

2419 

76 

14 

2419 

2436 

2453 

2470 

2487 

2504 

2521 

2538 

2554 

2571 

2588 

75 

15 

0.2588 

2605 

2622 

2639 

2656 

2672 

2689 

2706 

2723 

2740 

2756 

74 

16 

2756 

2773 

2790 

2807 

2823 

2840 

2857 

2874 

2890 

2907 

2924 

73 

17 

2924 

2940 

2957 

2974 

2990 

3007 

3024 

3040 

3057 

3074 

3090 

72 

18 

3090 

3107 

3123 

3140 

3156 

3173 

3190 

3206 

3223 

3239 

3256 

71 

19 

3256 

3272 

3289 

3305 

3322 

3338 

3355 

3371 

338? 

3404 

3420 

70 

20 

0.3420 

3437 

3453 

3469 

3486 

3502 

3518 

3535 

3551 

3567 

3584 

69 

21 

3584 

3600 

3616 

3633 

3649 

3665 

3681 

3697 

3714 

3730 

3746 

68 

22 

3746 

3762 

3778 

3795 

3811 

3827 

3843 

3859 

3875 

3891 

3907 

67 

23 

3907 

3923 

3939 

3955 

3971 

3987 

4003 

4019 

4035 

4051 

4067 

66  16 

24 

4067 

4083 

4099 

4115 

4131 

4147 

4163 

4179 

4195 

4210 

4226 

65 

25 

0.4226 

4242 

4258 

4274 

4289 

4305 

4321 

4337 

4352 

4368 

4384 

64 

26 

4384 

4399 

4415 

4431 

4446 

4462 

4478 

4493 

4509 

4524 

4540 

63 

27 

4540 

4555 

4571 

4586 

4602 

4617 

4633 

4648 

4664 

4679 

4695 

62 

28 

4695 

4710 

4726 

4741 

4756 

4772 

4787 

4802 

4818 

4833 

4848 

61 

29 

4848 

4863 

4879 

4894 

4909 

4924 

4939 

4955 

4970 

4985 

5000 

60 

30 

0.5000 

5015 

5030 

5045 

5060 

5075 

5090 

5105 

5120 

5135 

5150 

59  15 

31 

5150 

5165 

5180 

5195 

5210 

5225 

5240 

5255 

5270 

5284 

5299 

58 

32 

5299 

5314 

5329 

5344 

5358 

5373 

5388 

5402 

5417 

5432 

5446 

57 

33 

5446 

5461 

5476 

5490 

5505 

5519 

5534 

5548 

5563 

5577 

5592 

56 

34 

5592 

5606 

5621 

5635 

5650 

5664 

5678 

5693 

5707 

5721 

5736 

55 

35 

0.5736 

5750 

5764 

5779 

5793 

5807 

5821 

5835 

5850 

5864 

5878 

54 

36 

5878 

5892 

5906 

5920 

59a4 

5948 

5962 

5976 

5990 

6004 

6018 

53  14 

37 

6018 

6032 

6046 

6060 

6074 

6088 

6101 

6115 

6129 

6143 

6157 

52 

38 

6157 

6170 

6184 

6198 

6211 

6225 

6239 

6252 

6266 

6280 

6293 

51 

39 

6293 

6307 

6320 

6334 

6347 

6361 

6374 

6388 

6401 

6414 

6428 

50 

40 

0.6428 

6441 

6455 

6468 

6481 

6494 

6508 

6521 

6534 

6547 

6561 

49 

41 

6561 

6574 

6587 

6600 

6613 

6626 

6639 

6652 

6665 

6678 

6691 

48  13 

42 

6691 

6704 

6717 

6730 

6743 

6756 

6769 

6782 

6794 

6807 

6820 

47 

43 

6820 

6833 

6845 

6858 

6871 

6884 

6896 

6909 

6921 

6934 

6947 

46 

44° 

6947 

6959 

6972 

6984 

6997 

7009 

7022 

7034 

7046 

7059 

7071 

45° 

Complement 

.9 

.8 

.7 

.6 

.5 

.4 

.3 

.2 

.1 

.0 

Angle 

NATURAL  COSINES 


232 


APPENDIX 


NATURAL  SINES 


n    Complement 

Angle 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

Difference 

45° 

0.7071 

7083 

7096 

7108 

7120 

7133 

7145 

7157 

7169 

7181 

7193 

44° 

46 

7193 

T206 

7218 

7230 

7242 

7254 

7266 

7278 

7290 

7302 

7314 

43  12 

47 

7314 

7325 

7337 

7349 

7361 

7373 

7385 

7396 

7408 

7420 

7431 

42 

48 

7431 

7443 

7455 

7466 

7478 

7490 

7501 

7513 

7524 

7536 

7547 

41 

49 

7547 

7559 

7570 

7581 

7593 

7604 

7615 

7627 

7638 

7649 

7660 

40 

50 

0.7660 

7672 

7683 

7694 

7705 

7716 

7727 

7738 

7749 

7760 

7771 

39 

51 

7771 

7782 

7793 

7804 

7815 

7826 

7837 

7848 

7859 

7869 

7880 

38  " 

52 

7880 

7891 

7902 

7912 

7923 

7934 

7944 

7955 

7965 

7976 

7986 

37 

63 

7986 

7997 

8007 

8018 

8028 

8039 

8049 

8059 

8070 

8080 

8090 

36 

54 

8090 

8100 

8111 

8121 

8131 

8141 

8151 

8161 

8171 

8181 

8192 

35 

55 

0.8192 

8202 

8211 

8221 

8231 

8241 

8251 

8261 

8271 

8281 

8290 

34  10 

56 

8290 

8300 

8310 

8320 

8329 

8339 

8348 

8358 

8368 

8377 

838^3 

33 

57 

8387 

8396 

8406 

8415 

8425 

8434 

8443 

8453 

8462 

8471 

8480 

32 

58 

8480 

8490 

8499 

8508 

8517 

8526 

8536 

8545 

8554 

8563 

8572 

31 

59 

8572 

8581 

8590 

8599 

8607 

8616 

8625 

8634 

8643 

8652 

8660 

30  9 

60 

0.8660 

8669 

8678 

8686 

8695 

8704 

8712 

8721 

8729 

8738 

8746 

29 

61 

8746 

8755 

8763 

8771 

8780 

8788 

8796 

8805 

8813 

8821 

8829 

28 

62 

8829 

8838 

8846 

8854 

8862 

8870 

8878 

8886 

8894 

8902 

8910 

27  8 

68 

8910 

8918 

8926 

8934 

8942 

8949 

8957 

8965 

8973 

8980 

8988 

26 

64 

8988 

8996 

9003 

9011 

9018 

9026 

9033 

9041 

9048 

'9056 

9063 

25 

65 

0.9063 

9070 

9078 

9085 

9092 

9100 

9107 

9114 

9121 

9128 

9135 

24 

66 

9135 

9143 

9150 

9157 

9164 

9171 

9178 

9184 

9191 

9198 

9205 

23  7 

67 

9205 

9212 

9219 

9225 

9232 

9239 

9245 

9252 

9259 

9265 

9272 

22 

68 

9272 

9278 

9285 

9291 

9298 

9304 

9311 

9317 

9323 

9330 

9336 

21 

69 

9336 

9342 

9348 

9354 

9361 

9367 

9373 

9379 

9385 

9391 

9397 

20  6 

70 

0.9397 

9403 

9409 

9415 

9421 

9426 

9432 

9438 

9444 

9449 

9455 

19 

71 

9455 

9461 

9466 

9472 

9478 

9483 

9489 

9494 

9500 

9505 

9511 

18 

72 

9511 

9516 

9521 

9527 

9532 

9537 

9542 

9548 

9553 

9558 

9563 

17 

73 

9563 

9568 

9573 

9578 

9583 

9588 

9593 

9598 

9603 

-  9608 

9613 

16  5 

74 

9613 

9617 

9622 

9627 

9632 

9636 

9641 

9646 

9650 

9655 

9659 

15 

75 

0.9659 

9664 

9668 

9673 

9677 

9681 

9686 

9690 

9694 

9699 

9703 

14 

76 

9703 

9707 

9711 

9715 

9720 

9724 

9728 

9732 

9736 

9740 

9744 

13  * 

77 

9744 

9748 

9751 

9755 

9759 

9763 

9767 

9770 

9774 

9778 

9781 

12 

78 

9781 

9785 

9789 

9792 

9796 

9799 

9803 

9806 

9810 

9813 

1  9816 

11 

79 

9816 

9820 

9823 

9826 

9829 

9833 

9836 

9839 

9842 

9845 

9848 

10 

80 

0.9848 

9851 

9854 

9857 

9860 

9863 

9866 

9869 

9871 

9874 

9877 

9  3 

81 

9877 

9880 

9882 

9885 

9888 

9890 

9893 

9895 

9898 

9900 

9903 

8 

82 

9903 

9905 

9907 

9910 

9912 

9914 

9917 

9919 

9921 

9923 

9925 

7 

83 

9925 

9928 

9930 

9932 

9934 

9936 

9938 

9940 

9942 

9943 

9945 

6  2 

84 

9945 

9947 

9949 

9951 

9952 

9954 

9956 

9957 

9959 

9960 

9962 

5 

85 

0.9962 

9963 

9965 

9966 

9968 

9969 

9971 

9972 

9973 

9974 

9976 

4 

86 

9976 

9977 

9978 

9979 

9980 

9981 

9982 

9983 

9984 

9985 

9986 

3  l 

87 

9986 

9987 

9988 

9989 

9990 

9990 

9991 

9992 

9993 

9993 

9994 

2 

88 

9994 

9995 

9995 

9996 

9996 

9997 

0997 

9997 

9998 

9998 

9998 

1 

89° 

9998 

9999 

9999 

9999 

9999 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

0°° 

Complement 

.9 

.8 

.7 

.6 

.5 

.4 

.3 

.2 

.1 

.0 

Angle 

NATURAL  COSINES 


APPENDIX 


233 


NATURAL  TANGENTS 


Complement 

Angle 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

Difference 

0° 

0.0000 

0017 

0035 

0052 

0070 

0087 

0105 

0122 

0140 

0157 

0175 

89° 

I 

0175 

0192 

0209 

0227 

0244 

0262 

0279 

0297 

0314 

0332 

0349 

88 

2 

0349 

0367 

0384 

0402 

0419 

0437 

0454 

0472 

0489 

0507 

0524 

87 

3 

0524 

0542 

0559 

0577 

0594 

0612 

0629 

0647 

0664 

0682 

0699 

86 

4 

0699 

0717 

0734 

0752 

0769 

0787 

0805 

0822 

0840 

0857 

0875 

85 

5 

00875 

0892 

0910 

0928 

0945 

0963 

0981 

0998 

1016 

1033 

1051 

84 

6 

1051 

1069 

1086 

1104 

1122 

1139 

1157 

1175 

1192 

1210 

1228 

83 

7 

1228 

1246 

1263 

1281 

1299 

1317 

1334 

1352 

1370 

1388 

1405 

82 

8 

1405 

1423 

1441 

1459 

1477 

1495 

1512 

1530 

1548 

1566 

1584 

81 

9 

1584 

1602 

1620 

1638 

1655 

1673 

1691 

1709 

1727 

1745 

1763 

80 

10 

0.1763 

1781 

1799 

1817 

1835 

1853 

1871 

1890 

1908 

1926 

1944 

79  w 

11 

1944 

1962 

1980 

1998 

2016 

2035 

2053 

2071 

2089 

2107 

2126 

78 

12 

2126 

2144 

2162 

2180 

2199 

2217 

2235 

2254 

2272 

2290 

2309 

77 

13 

2309 

2327 

2345 

2364 

2382 

2401 

2419 

2438 

2456 

2475 

2493 

76 

14 

2493 

2512 

2530 

2549 

2568 

2586 

2605 

2623 

2642 

2661 

2679 

75 

15 

0.2679 

2698 

2717 

2736 

2754 

2774 

2792 

2811 

2830 

2849 

2867 

74 

16 

.  2867 

2886 

2905 

2924 

2943 

2962 

2981 

3000 

3019 

3038 

3057 

73  19 

17 

3057 

3076 

3096 

3115 

3134 

3153 

3172 

3191 

3211 

3230 

3249 

72 

18 

3249 

3269 

3288 

3307 

3327 

3346 

3365 

3385 

3404 

3424 

3443 

71 

19 

3443 

3463 

3482 

3502 

3522 

3541 

3561 

3581 

3600 

3620 

3640 

70 

20 

0.3640 

3659 

3679 

3699 

3719 

3739 

3759 

3779 

3799 

3819 

3839 

69 

21 

3839 

3859 

3879 

3899 

3919 

3939 

3959 

3979 

4000 

4020 

4040 

68  20 

22 

4040 

4061 

4081 

4101 

4122 

4142 

4163 

4183 

4204 

4224 

4245 

67 

23 

4245 

4265 

4286 

4307 

4327 

4348 

4369 

4390 

4411 

4431 

4452 

66 

24 

4452 

4473 

4494 

4515 

4536 

4557 

4578 

4599 

4621 

4642 

4663 

65  21 

25 

0.4663 

4684 

4706 

4727 

4748 

4770 

4791 

4813 

4834 

4856 

4877 

64 

26 

4877 

4899 

4921 

4942 

4964 

4986 

5008 

5029 

5051 

5073 

5095 

63 

27 

5095 

5117 

5139 

5161 

5184 

5206 

5228 

5250 

5272 

5295 

5317 

62  22 

28 

5317 

5340 

5362 

5384 

5407 

5430 

5452 

5475 

5498 

5520 

5543 

61 

29 

5543 

5566 

5589 

5612 

5635 

5658 

5681 

5704 

5727 

5750 

5774 

60  23 

30 

0.5774 

5797 

5820 

5844 

5867 

5890 

5914 

5938 

5961 

5985 

6099 

59 

31 

6009 

6032 

6056 

6080 

6104 

6128 

6152 

6176 

6200 

6224 

6249 

58  2* 

32 

6249 

6273 

6297 

6322 

6346 

6371 

6395 

6420 

6445 

6469 

6494 

57 

33 

6494 

6519 

6544 

6569 

6594 

6619 

6644 

6669 

6694 

6720 

6745 

56  25 

34 

6745 

6771 

6796 

6822 

6847 

6873 

6899 

6924 

6950 

6976 

7002 

55 

35 

0.7002 

7028 

7054 

7080 

7107 

7133 

7159 

7186 

7212 

7239 

7265 

54  26 

36 

7265 

7292 

7319 

7346 

7373 

7400 

7427 

7454 

7481 

7508 

7536 

53  27 

37 

7536 

7563 

7590 

7618 

7646 

7673 

7701 

7729 

7757 

7785 

7813 

52  28 

38 

7813 

7841 

7869 

7898 

7926 

7954 

7983 

8012 

8040 

8069 

8098 

51  28 

39 

8098 

8127 

8156 

8185 

8214 

8243 

8273 

8302 

8332 

8361 

8391 

50  29 

40 

0.8391 

8421 

8451 

8481 

8511 

8541 

8571 

8601 

8632 

8662 

8693 

49  30 

41 

8693 

8724 

8754 

8785 

8816 

8847 

8878 

8910 

8941 

8972 

9004 

48  31 

42 

9004 

9036 

9067 

9099 

9131 

9163 

9195 

9228 

9260 

9293 

9325 

47  32 

43 

9325 

9358 

9391 

9424 

9557 

9490 

9523 

9556 

9590 

9623 

9657 

46  33 

44° 

9657 

9691 

9725 

9759 

9793 

9827 

9861 

9896 

9930 

9965 

1.0000 

45°3* 

Complement 

.9 

.8 

.7  • 

.6 

.5 

.4 

.3 

.2 

.1 

.0 

Angle 

NATURAL  COTANGENTS 


234 


APPENDIX 


NATURAL  TANGENTS 


Angle 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

Dif. 

45° 

1.0000 

1.0035 

1.0070 

1.0105 

1.0141 

1.0176 

1.0212 

1.0247 

1.0283 

1.0319 

36 

46 

1.0355 

1.0392 

1.0428 

1.0464 

1.0501 

1.0538 

1.0575 

1.0612 

1.0649 

1.0686 

37 

47 

1.0724 

1.0761 

1.0799 

1.0837 

1.0875 

1.0913 

1.0951 

1.0990 

1.1028 

1.1067 

88 

48 

1.1106 

1.1145 

1.1184 

1.1224 

1.1263 

1.1308 

1.1343 

1.1383 

1.  1423 

1.1463 

40 

49 

1.  1504 

1.1544 

1.1585 

1.1626 

1.1667 

1.1708 

1.1750 

1.1792 

1.1833 

1.1875 

41 

50 

1.1918 

1.1960 

1.2002 

1.2045 

1.2088 

1.2131 

1.2174 

1.2218 

1.2261 

1.2305 

43 

51 

1.2849 

1.2393 

1.243" 

1.2482 

1.2527 

1.2572 

1.2617 

1.2662 

1.2708 

1.2753 

45 

52 

1.2799 

1.2846 

1.2892 

1.2938 

1.2985 

1.3032 

1.3079 

1.3127 

1.3175 

1.3222 

47 

53 

1.3270 

1.3319 

1.3367 

1.3416 

1.3465 

1.3514 

1.3564 

1.3613 

1.3663 

1.3713 

49 

54 

1.3764 

1.3814 

1.3865 

1.3916 

1.3968 

1.4019 

1.4071 

1.4124 

1.4176 

1.4229 

52 

55 

1.4281 

1.4335 

1.4388 

1.4442 

1.4496 

1.4550 

1.4605 

1.4659 

1.4715 

1.4770 

54 

56 

1.4826 

1.4882 

1.4938 

1.4994 

1.5051 

1.5108 

1.5166 

1.52M 

1.5282 

1.5340 

57 

57 

1.5399 

1.5458 

1.5517 

1.5577 

1.5637 

1.5697 

1.5757 

1.5818 

1.5880 

1.5941 

60 

58 

1.6003 

1.6066 

1.6128 

1.6191 

1.6255 

1.6319 

1.6383 

1.6447 

1.6512 

1.6577 

64 

59 

1.6643 

1.6709 

1.6775 

1.6842 

1.6909 

1.6977 

1.7045 

1.7113 

1.7182 

1.7251 

68 

60 

1.7321 

1.7391 

1.7461 

1.7532 

1.7603 

1.7675 

1.7747 

1.7820 

1.7893 

1.7966 

72 

61 

1.8040 

1.8115 

1.8190 

1.8265 

1.8341 

1.8418 

1.8495 

1.8572 

1.8650 

1.8728 

77 

62 

1.880r/ 

1.8887 

1.8967 

1.9047 

1.9128 

1.9210 

1.9292 

1.93751.9458 

1.9542 

82 

63 

1.9626 

1.9711 

1.9797 

1.9883 

1.9970 

2.0057 

2.0145 

2.0233 

2.0323 

2.0413 

88 

64 

2.0503 

2.0594 

2.0686 

2.0778 

2.0872 

2.0965 

2.1060 

2.1155 

2.1251 

2.1848 

94 

65 

2.145 

2.154 

2.164 

2.174 

2.184 

2.194 

2.204 

2.215 

2.225 

2.236 

10 

66 

2.246 

2.257 

2.267 

2.278 

2.289 

2.300 

2.311 

2.322 

2.333 

2,344 

11 

67 

2.356 

2.367 

2.379 

2.391 

2.402 

2.414 

2.426 

2.438 

2.450 

2.463 

12 

68 

2.475 

2.488 

2.500 

2.513 

2.526 

2.539 

2.552 

2.565 

2.578 

2.592 

13 

69 

2.605 

2.619 

2.633 

2.646 

2.660 

2.675 

2.689 

2.703 

'2.718 

2.733 

14 

70 

2.747 

2.762 

2.778 

2.793 

2.808 

2.824 

2.840 

2.856 

2.872 

2.888 

16 

71 

2.904 

2.921 

2.937 

2.954 

2.971 

2,989 

1006 

3.024 

3.042 

3.060 

17 

72 

3.078 

3.096 

3.115 

3.133 

3.152 

3.172 

3.191 

3.211 

3.230 

3.250 

19 

73 

3.271 

3.291 

3.312 

8.333 

3.354 

3.376 

3.398 

3.420 

3.442 

3.465 

22 

74 

3.487 

3.511 

3.534 

3.558 

3.582 

3.606 

3.630 

3.655 

3.681 

3.700 

25 

75 

3.732 

3.758 

3.785 

3.812 

3.839 

3.867 

3.895 

3.923 

3.952 

3.981 

28 

76 

4.011 

4.041 

4.071 

4.102 

4.134 

4.165 

4.198 

4.230 

4.264 

4.297 

32 

77 

4.331 

4.366 

4.402 

4.437 

4.474 

4.511 

4.548 

4.586 

4.625 

4.665 

37 

78 

4.705 

4.745 

4.787 

4.829 

4.872 

4.915 

4.959 

5.005 

5.050 

5.097 

44 

79 

5.145 

5.193 

5.242 

5.292 

5.343 

5.396 

5.449 

5.503 

5.558 

5.614 

52 

80 

5.67 

5.73 

5.79 

5.85 

5.91 

5.98 

6.04 

6.11 

6.17 

6.24 

7 

81 

6.31 

6.39 

6.46 

6.54 

6.61 

6.69 

6.77 

6.85 

6.94 

7.03 

8 

82 

7.12 

7.21 

7.30 

7.40 

7.49 

7.60 

7.70 

7.81 

7.92 

8.03 

10 

83 

8.14 

8.26 

8.39 

8.51 

8.64 

8.78 

8.92 

9.06 

9.21 

9.36 

14 

84 

9.51 

9.68 

9.84 

10.0 

10.2 

10.4 

10.6 

10.8 

11.0 

11.2 

85 

11.4 

11.7 

11.9 

12.2 

12.4 

12.7 

13.0 

13.3 

13.6 

14.0 

3 

86 

14.3 

14.7 

15.1 

15.5 

15.9 

16.3 

16.8 

17.3 

17.9 

18.5 

6 

87 

19.1 

19.7 

20.4 

21.2 

22.0 

22.9 

23.9 

24.9 

26.0 

27.3 

88 

28.6 

30.1 

31.8 

33.7 

35.8 

38.2 

40.9 

44.1 

47.7 

52.1 

89° 

57. 

64. 

72. 

82.       , 

95. 

115. 

43. 

191. 

286. 

573. 

Angle 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

NATURAL  TANGENTS 


•».*. 


APPENDIX 


LOGARITHMS 


235 


0 

1     1     2 

3 

4 

5 

0212 

6 

7 

8 

9 

1    2   3 

456 

789 

10 

0000 

0043|0086 

012£ 

50170 

0253 

0294 

0334 

0374 

4  8  12 

17  21  25 

29  33  37 

11 

12 
13 

14 
15 
16 

0414 
0792 

1461 
1761 

0453 

082S 
1173 

I 

0492 

0864 
1206 

Ii523 

1 

053l|0569 

08990934 
12391271 

15531584 

18471875 
21222148 

0607 
0969 
1303 

1614 
1903 
2175 

0645 
1004 
1335 

1644 
1931 
2201 

0682 
1038 
1367 

1673 
1959 

2227 

0719 
1072 

I3£y 

1703 
19"87 
2253 

0755 
1*06 
1430 

1732 
2014 
2279 

4-  8  11 
3  7  10 
3  6  10 

369 
368 
358 

15  19  23 
14  17  21 
13  16  19 

12  15  18 
11  14  17 
11  13  16 

26  30  34 
24  28  31 
23  26  29 

21  24  27 
20  22  25 
18  21  24 

17 

18 
19 

£304 
•,553 

a?88 

gD^Kj 

2810J3833 

2380Mtf  2430 
2625^02672 
2856^82900 

2455 
269o 
2923 

2480 
2718 
2945 

2504 
2742 
2967 

2529 
2765 

2989 

257 
257 
247 

10  12  15 
9  12  14 
9  11  13 

17  20  22 
16  19  21 
16  18  20 

20 

3010 

3032 

3054 

307530963118 

3139 

3160 

3181 

3201 

246 

8  11  13 

15  17  19 

21 
22 
23 

3222 
3424 
3617 

3243 
3444 
3636 

3263 
3464 
3655 

35^433043324 
348335023522 
3674|3692|3711 

3345 
3541 
3729 

3365 
3560 
3747 

3385 
3579 
3766 

3404 
3598 

3784 

246 
246 
246 

8  10  12 
8  10  12 
7    9  11 

14  16  18 
14  15  17 
13  15  17 

24 
25 
26 

27 

28 
29 

3802 
31)79 

"*' 

43U 
4472 
4624 

3820 
3997 
416 

433 

448 
46S 

3838 
4014 
4183 

4346 
4502 
4654 

385C 
4031 
420C 

4362 
466 

3874 
4048 
4216 

4378 
4533 
4683 

3892 
4065 
4232 

4393 

4548 
4698 

3909 
4082 
4249 

4409 
4564 
4713 

3927 
4099 
4265 

4425 
4579 

4728 

3945 
4116 

4281 

4440 
4594 

4742 

3962 
4133 
4298 

4456 
4609 

4757 

245 
235 
235 

235 
235 

1  3    4 

7    9  11 
7    9  10 

7    8  10 

689 
689 
679 

12  14  16 
12  14  15 
11  13  15 

11  13  14 
11  12  14 
10  12  13 

30 

4771 

478 

4800 

481 

4829 

4843 

4857 

4871 

488$ 

4900 

1  3    4 

679 

10  11  13 

31 
32 
33 

4914 
5051 

5185 

492 
506 
519 

4942 
5079 
5211 

495 
509 
522 

4969 
5105 
5237 

4983 
5119 
5250 

4997 
5132 
5263 

5011 
5145 
5276 

5024 
5159 
5289 

5038 
5172 
5302 

1  8    4 
1  3    4 
1  3    4 

678 
578 
568 

10  11  12 
9  11  12 

9  10  12 

34 
35 
36 

5315 
5441 
5563 

532 
545 
557 

5340 
5465 

5587 

535 
547 
559 

5366 
5490 
5611 

5378 
5502 
5623 

5391 
5514 
5635 

5403 
5527 
5647 

5416 
5539 

56-58 

5428 
5551 
5670 

1  3    4 
1  2    4 

1  2    4 

568 
567 
567 

9  10  11 
9  10  11 
8  10  11 

37 

38 
39 

5682 

5798 
5911 

569 
580 

592 

5705 
5821 
5933 

571 
583 
5944 

5729 
5843 
5955 

5740 
5855 
5966 

5752 
5866 
5977 

5763 

5877 
5988 

57f5 

5888 
5999 

5786 
5899 
6010 

1  2    3 
1  2    3 
1  2    3 

567 
567 
457 

8    9  10 
8     9  10 

8    9  10 

40 

6021 

603 

6042 

605 

6064 

6075 

6085 

6096 

6107 

6117 

1  2    3 

456 

8    9  10 

41 
42 
43 

6128 
6232 
6335 

613 
624 
634 

6149 
6253 
6355 

616 
626 
636 

6170 
6274 
6375 

6180 
6284 
6385 

6191 
6294 
6395 

6201 
6304 
6405 

6212 
6314 
6415 

6222 
6325 
6425 

123 
123 
1  2    3 

456 
456 
456 

789 
789 
789 

44 
45 
46 

6435 
6532 

6628 

644^ 
654 
663 

6454 
6551 
6646 

646 
656 
665 

6474 
6571 
6665 

6484 
6580 
6675 

6493 
6590 
6684 

6503 
6599 
6693 

6513 
6009 
6702 

6522 
6618 
6712 

123 
1  2    3 
L  2    3 

456 
456 
456 

789 
789 

778 

47 

48 
49 

6721 
6812 
6902 

673 

682 
691 

6739 
6830 
6920 

674 

683 
692 

6758 
6848 
6937 

6767 

6857 
6946 

6776 
6866 
6955 

6785 
6875 
6964 

6794 

6884 
6972 

6803 
6893 
6981 

1  2    3 
123 
1  2    3 

455 

445 
445 

678 
678 
678 

50 

6990 

699 

7007 

701 

7024 

7033 

7042 

7050 

7059 

7067 

1  2    3 

345 

678 

51 
52 
53 

7076 
7160 
7243 

7084 
7168 
7251 

7093 

7177 
7259 

710 
718 
726 

7110 
7193 

7275 

7118 
7202 

7284 

7126 
7210 
72£2 

7135 

7218 
7300 

7143 
7226 

7308 

7152 
7235 
7316 

1  2    3 
1  2    2, 
122 

345 
3    4    5 
345 

678 
677 
667 

54 

7324 

7332 

7340 

734 

7356 

7364 

7372 

7380 

7388 

7396 

1  2    2 

345 

667 

fei 


236 


APPENDIX 


LOGARITHMS 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1  2  3 

456789 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

122 

345567 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

122 

345567 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

122 

3  4  5L5  6  7 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

1  1  2 

3  4  4m  6  7 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

1  l^jW  -1  4r~5  6  7 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

1  iBt.  !  4566 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7963 

7910 

7917 

1  •  W  4  5  6  6 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

79§A&87 

1  •  R  4  5  6  6 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

<S(  )•».").-, 

1  1WG3  4|  5  5  6 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

81160*122 

112 

334556 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

112 

334556 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

112 

334556 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

1  1  2 

3  3  4l  5  5  6 

68 

8325 

8331 

8838 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

112 

334 

456 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

112 

234 

456 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

112 

234 

456 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

112 

23| 

455 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

1   2 

234 

455 

8633 

8639 

8645 

8651 

8657 

86C& 

8669 

8675 

8681 

§686 

1   2 

234 

455 

?i 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

1   2 

«  3  4 

455 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

1   2 

233 

455 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

1   2 

233 

4  5  5 

77 

8865 

8871 

8876 

8882 

8887 

8893 

88*99 

8904 

8910 

8915 

1   2 

233 

4  4  5 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

1   2 

233 

445 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

1  1  2 

233 

4  4  5 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

112 

233 

445 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

112 

233 

445 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

112 

233 

445 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

112 

233 

445 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

1  1  2 

233 

445 

85 

.9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

1  1  2 

233 

445 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

112 

233 

445 

87 

9395 

9400 

9405 

9410 

94  1  5 

9420 

9425 

9430 

9435 

9440 

0  1  1 

223 

344 

88 

9445 

9450 

9455 

94-60 

9465 

9469 

9474 

9479 

9484 

9489 

0  1  1 

223 

344 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

0  1  1 

223 

344 

90 

9542 

9547 

9552 

9557 

9562 

•9566 

9571 

9576 

9581 

9586 

0  1  1 

223 

344 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

0  1  1 

223 

344 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

0  1  1 

223 

344 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

0  1  1 

223 

344 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

97681  9773 

Oil 

223 

344 

95 

9777 

9782 

9786 

,9791 

9795 

9800 

9805 

9809 

9814  9818 

Oil 

223 

344 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859  9863 

0  1  1 

223 

344 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903  9908 

0  1  1 

223 

344 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948|  9952 

0  1  1 

223 

344 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

Oil 

223 

334 

INDEX 


Absolute  expansion  of  mercury,  218. 

humidity,  164. 

temperature,  136,  145. 

zero,  128. 
Acceleration,  angular,  78,  85. 

linear,  9. 

uniform,  9. 

Acceleration  machine,  11. 
Adjustment  of  cathetometer,  109. 
Air,  correction  in  weighing,  173. 

table  of  density  of  dry,  227. 
Alluard,    form  of  hygrometer,   167, 

170. 

Apparent  expansion,  216. 
Archimedes'  principle,  172. 
Aspirator,  160. 
Atomic  theory,  138. 

weights,  144. 

August,    wet-and-dry  bulb   hygrom- 
eter, 169. 
Avogadro,  law  of,  131,  138,  143. 

Balance,  analytical,  116. 

Mohr's,  175. 

simple,  36,  37,  38. 

spring,  26. 
Barometer,  108. 

corrections,  table  of,  229. 
Bernouilli,  work  on  the  kinetic  the- 
ory, 106. 

Black,  theory  of  heat,  198. 
Blow-pipe,  149. 

Boiling  temperature.  133,  156. 
Boyle's  Law,  105. 

departures  from,  107. 
Brake,  Prony,  48. 
Brunner,  the  chemical  hygrometer, 

168. 

Bunsen  ice  calorimeter,  208. 
Buoyancy,  correction  for,  173 


Cailletet,      liquefaction     of     gases 

128 
Caliper,  micrometer,  223. 

vernier,  224: 
Caloric,  198. 

Calorie,  definition  of,  198,  203. 
Calorimeter,  ice,  208 

steam,  208 

Calorimetric  methods,  204. 
Calorimetry,  198. 

Capillary  constant,  measurement  of, 
186,  194. 

correction,  159. 

depressions,  table  of,  229. 

phenomena,  182. 

tubes,  change  of  level  in,  185. 
Carbon  dioxide,  solid,  128. 
Cathetometer,  109. 
Center  of  gravity,  34,  35,  37. 

of  inertia,  33,  35. 

of  mass,  definition  of,  33,  34. 

of  oscillation,  .96. 
Centripetal  force,  100. 
Chappuis,     determinations     of     gas 

coefficients,  126. 
Charles's  Law,  122. 
Chemical  hygrometer,  168. 
Circular  motion,  uniform,  101. 
Clausius,    development     of    the    ki- 
netic theory,  106. 

Coefficient  of  expansion,   definition 
of,  215. 

of  solids  and  liquids,  216. 

of  gases,  124. 

of  glass,  133. 

table  of,  230. 
Coefficient  of  restitution,  58,  59. 

of  rigidity,  definition  of,  72. 

of  elasticity,  table  of,  226. 
Coincidences,  method  of,  96. 


237 


238 


IXDEX 


Component  of  a  force,  definition  of, 

23. 
Composition      and      resolution     of 

forces,  21. 

Compound  pendulum,  95. 
Conservation  of  energy,  45. 
Constant  proportions,  law  of,  138. 
Constant  of  radiation,  204. 
Constant-volume  thermometer,   125. 
Contact  angle,  187. 
Contractility  of  a  surface  film,  192. 
Cooling,  curves  of,  210., 

by  evaporation,  155. 

law  of,  205. 

method  of,  206. 

Correction  of  a  mercury  thermome- 
ter, 135. 
Curve  of  saturated  vapor,  152. 

Dalton,  development  of  the  atomic 
theory,  138. 

law     of     multiple    proportions, 

140. 

Daniell,  dew-point  hygrometer,  166. 
Density,  of  air,  114, 

of  dry  air,  table  of,  227. 

of  carbon  dioxide,  145. 

of  gases  and  vapors,  138. 

of  gases,  table  of,  144. 

of  a  liquid,  174. 

of  mercury,  table  of,  227. 

of  a  solid,  174. 

of  solids,  liquids,  and  gases,  table 
of,  226. 

of  water,  table  of,  227. 
Dewar,  liquefaction  of  gases,  128. 
Dew-point,  165. 

Dew-point  hygrometer,  166,  167. 
Dimensional  formulae,  84,  85. 
Disk,  inertia,  82. 

Donny,  superheating  of  water,  157. 
Double  weighing,  39,  173. 
Dufour,  superheating  of  water,  157. 
Dulong  and  Petit,  law  of,  202. 

method  for  expansion  of    mer- 
cury, 218. 
Dyne,  definition  of,  17. 


Efficiency,  definition  of,  46. 
Elasticity,  65. 

coefficients  of,  65. 

table  of,  226. 
Energy,  definition  of,  4£ 

conservation  of,  45 

and  efficiency,  42 

kinetic,  42,  43. 

molecular,  129 

potential,  44. 

of  rotation,  84 

unit  of,  42. 

Equation  of  moments,  32,  37. 
Equili  brant,  33 

Equivalence,  principle  of,  139. 
Evaporation,  155. 

Expansion,    apparent  and   absolute, 
216 

of  mercury,  absolute,  218. 

coefficient  of,  215 

coefficient  of  gases,  123,  126,  215. 

coefficients  of  solids  and  liquids, 
215,  218. 

coefficients,  table  of,  230. 


Faraday,      liquefaction     of      gases, 

128. 

Field  of  force,  uniform,  34. 
Films,  theory  of  thin  liquid,  189. 
Force,  13,  15 

absolute  unit  of,  17. 

centripetal,  100. 

composition  and  resolution  of,  21, 
22,  23. 

constant  of  a  spring,  90. 

definition  of,  17. 

gravitational  unit  of,  17. 

inter  molecular,   182. 

table,  26. 
Freezing  mixture,  135. 


"g,     determination  of,  95. 

"gr,"  value  of,  13 

Galileo's  definition  of  force,  15. 

Gases,  theory  of,  106. 

Gas  thermometers,  123. 


INDEX 


239 


Gay-Lussac,  law  of,  122. 

experiments  on  temperature  of 

water  and  its  vapor,  157. 
Glass,   coefficient   of   expansion  of, 

133. 

Gram,  definition  of,  17. 
Gram -molecule.  201. 
Gravitation,  law  of,  20. 
Gravity,  center  of,  34,  35,  37. 
Gyration,  radius  of,  96. 

Heat  capacity,  definition  of,  200. 

latent,  199. 

table  of  latent,  200. 

mechanical  equivalent  of,  45. 

molecular,  201. 

table  of  molecular,  201. 

definition  of  specific,  200. 

table  of  specific,  201,  230. 

early  theory  of,  198. 
Heats,  specific,  variation  with  tem- 
perature, 203. 
Henry,    measurement    of    capillary 

constant,  196. 
Hooke's  Law,  66,  91. 
Horse -power,  51. 
Humidity,  absolute,  164. 

relative,  164. 
Hydrogen,  liquid,  128. 

solid,  128. 

thermometer,  127. 
Hydrometer,  constant-weight,  178. 
Hygrometer,  absorption,  165. 

chemical,  168. 

dew-point,  166,  167. 

wet-and-dry  bulb,  169. 
Hygrometry,  164. 

Ice  calorimeter,  208. 
Ice,  latent  heat  of,  199. 
Impact,  elastic,  61. 

inelastic,  54. 

laws  of,  52,  53. 
Inertia,  78. 

center  of,  33,  35. 

moment  of,  78,  79. 
Inertia  disk,  82. 


International  Bureau  of  Weights  and 
Measures,  method  of  meas- 
uring linear  expansion  coeffi- 
cients, 219. 

Johonnott,  thickness    of    the    black 

spot,  191. 
Joule,  equivalent  of  heat,  45,  46. 

development  of  the  kinetic  the- 
ory, 106. 

relation    between   pressure  and 
molecular  energy  of  a  gas, 
129. 
definition  of,  51. 

Kinetic  energy,  42,  43. 

energy  in  impact,  loss  of,  53,  59. 
energy  of  rotation,  84. 
energy  of  translation,  84. 
theory  of  gases,  106. 
theory  of  liquids,  152. 
theory  and  temperature,  127. 

Laplace,  formula  and  its  deduction, 

193. 
theory    of    molecular    pressure, 

183. 

normal  pressure,  185. 
Latent  heat,  199. 
table  of,  200. 

Lavoisier  and  Laplace,  linear  coeffi- 
cient, 219. 

LeRoy,   dew-point  hygrometer,   166. 
Leslie,    wet-and-dry    bulb    hygrom- 
eter, 169. 

Lever  arm,  definition  of,  32. 
Lever,  law  of,  31,  32. 

optical,  68. 

Linear  expansion  coefficient,  218. 
Liquid,  density  of,  174. 
Logarithms,  table  of,  235. 

Manometer,  48,  49. 
Mariotte's  Law,  105. 
Mass,  unit  of,  17. 

gravitational  unit  of,  17. 

definition  of  center  of,  33,  34. 
Maxwell,  proof  of  Avogadro's  Law, 
145. 


240 


INDEX 


Mayer,  conservation  of  energy,  45. 
Mean  free  path,  153. 
Mechanical  equivalent  of  heat,  45. 
Mercury,  table  of  densities  of,  227. 
Method  of  coincidences,  96 

of  cooling,  206. 

of  mixture,  204,  211. 

of  oscillation,  117. 
Micrometer  caliper,  223. 
Modulus  of  torsion,  74. 
Modulus,  volume,  66. 

Young's,  67. 
Mohr's  balance,  175. 
Molecular  energy  and  pressure  of  a 
gas,  129. 

force,  182. 

heat,  201. 

heats,  table  of,  201. 

pressure,  183. 

pressure,  relation  to  surface  ten- 
sion, 192. 

pressure,  variation  with  curva- 
ture, 184. 

velocity,  100,  130. 

weights,  144,  201. 
Molecule,  139,  152. 
Moment  of  force,  definition  of,  32. 

of  inertia,  definition  of,  78,  79 

of  torsion,  definition  of,  75. 
Moments,  equation  of,  32,  37. 

principle  of,  30,  31. 
Momentum,  15,  43. 

conservation  of,  52,  53. 
Motion,  rotary,  81. 

simple  harmonic,  87. 

uniformly  accelerated,  9. 

uniform  circular,  101. 
Motor,  water,  48. 
Multiple  proportions,  law  of,  140. 

Neumann,  law  of,  202. 
Newton,  interpretation  of  Third  Law, 
42. 

coefficient  of  restitution,  58. 

conservation  of  energy,  45. 

law  of  cooling,  205,  211. 

law  of  gravitation,  20. 


Newton,  laws  of  motion,  15,  17. 
Nitrous  oxide,  liquid,  128. 
Normal  pressure,  185. 

Olszewski,  liquefaction  of  gases,  128. 
Optical  lever,  68. 
Oscillation,  center  of,  96. 

method  of,  117. 
Oxygen,  liquefaction  of,  128. 

Parallax,  error  of,  12. 
Parallelogram  of  forces,  22. 
Pascal's  Law,  181 
Pendulum,  compound,  95. 

simple,  96. 

torsion,  92. 

Pictet,  liquefaction  of  gases,  128. 
Potential  energy,  44. 
Pressure  coefficient  of  gases,  125,  126. 

exerted  by  a  gas,  129,  130. 

law  of  liquid,  181. 
Prony  brake,  48. 
Pulleys,  system  of,  47. 
Pump,  water,  160. 

Radian,  68,  75,  79. 
Radiation  constant,  204. 
Radius  of  gyration,  96. 

of  influence  of  molecular  force, 

183. 
Rayleigh,  measurement  of  thickness 

of  oil  films,  190. 

Reduction  of  vapor  pressures,  159. 
Regnault,    experiments    on    Boyle's 

Law,  107. 

device  of  counterpoise  in  weigh- 
ing, 115. 
experiments     on     hygrometers, 

166. 

coefficient  of  expansion  of  mer- 
cury, 218. 

Relative  humidity,  164. 
Repulsion  theory  of  gases,  105. 
Resolution  of  forces,  21. 
Resultant,  definition  of,  22. 
of  forces,  23,  24. 


INDEX 


241 


Restitution,  coefficient  of,  58,  59,  61. 
Rigidity,  coefficient  of,  65,  71,  72. 
Rotary  motion,  81. 

Rowland,    experiments    on    specific 
heat  of  water,  203. 

Sanctorius,    absorption  hygrometer, 

165. 

Saturated  vapor,  154. 
Saussure,  absorption  hygrometer,  165. 
Scholium  to  third  law  of  motion,  29. 
Sensitiveness  of  a  balance,  118,  177. 
Simple  harmonic  motion,  87,  90. 

pendulum,  96. 

Specific  heat,  definition  of,  200. 
table  of,  201,  230. 
variation  with  temperature,  203 
Spring  balance,  26. 
Steam  calorimeter,  208. 
latent  heat  of,  199. 
Stevin,  proof  of  Archimedes'  princi 

pie,  172. 

Substitution,  principle  of,  141. 
Superheating  of  water,  157. 
Surface  tension,  156,  190. 
measurement  of,  195. 
relation  to  molecular  pressure 

192. 

table  of,  230. 
theory  of,  190. 


Table    of    capillary    depressions    o 

mercury,  229. 

of  coefficients  of  expansion,  23 
of  densities  of  dry  air,  227. 
of  densities  of  mercury,  227. 
of  densities  of  solids,  liquids,  an 

gases,  226. 
of    densities    and    tensions 

water  vapor,  228. 
of  densities  of  water,  227. 
of  elastic  coefficients,  226. 
of  logarithms,  235. 
of     reductions     of     baromet 

heights,  229. 
of  specific  heats,  230. 


able  of  surface  tensions,  230. 

of  trigonometric  functions,  231. 
emperature,  absolute,  136,  145. 

boiling,  156. 

measurement  of,  122. 
"ension  in  a  cord,  19. 

surface,  156,  190. 

measurement  of  surface,  19."). 

theory  of  surface,  '190. 

table  of  surface,  230. 

of  saturated  vapor,  158,  165. 
hermometer,  gas,  123. 

corrections,  135,  158. 

corrections,  curve  of,  159. 

hydrogen,  127. 
orsion,  modulus  of,  74. 

moment  of,  75. 

pendulum,  92. 
Trigonometric    functions,   table  of> 
231. 

Uniform  accelerated  motion,  9,  10. 
circular  motion,  101. 
field  of  force,  34. 

Vapor  pressures,  reduction  of,  159. 
saturated,  154. 
tension,  158. 
tension,  table  of,  228. 
Vaporization,  153. 

Variation  of  specific  heats  with  tem- 
perature, 203. 
Vector,  100. 
Velocity,  21 
angular,  84. 
mean,  10. 

molecular,  106,  130. 
Vernier,  224. 

caliper,  225. 

Vinci,  capillary  phenomena,  182. 
Violle,  specific  heat  of  platinum,  213. 
Volume  coefficients  of  solids,  217. 
modulus,  66. 


Water  density,  table  of,  227. 
motor,  48. 
pump,  160. 


242 


INDEX 


Watt,  definition  of,  51. 

Weighing,  method  of  double,  39,  173. 

with  false  balances,  38. 
Weight  of  body  in  vacuo,  172. 
Weights,  atomic  and  molecular,  144. 
Wet-and-dry  bulb  hygrometer,  169. 
Woestyn,  law  of,  202. 
Work,  definition  of,  29. 


Work,  principle  of,  29. 

unit  of,  42. 
Wroblewski,    liquefaction  of 

128. 

Young's  modulus,  67. 
Zero,  absolute,  128. 


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